Results of Bessel Functions I: Difference between revisions

Latest revision as of 06:59, 25 May 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.2.E1	${\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+% z\frac{\mathrm{d}w}{\mathrm{d}z}+(z^{2}-\nu^{2})w=0}}$ `z^{2}\deriv[2]{w}{z}+z\deriv{w}{z}+(z^{2}-\nu^{2})w = 0`		(z)^(2)* diff(w, [z$(2)])+ zdiff(w, z)+((z)^(2)- (nu)^(2))w = 0	(z)^(2)* D[w, {z, 2}]+ zD[w, z]+((z)^(2)- \[Nu]^(2))w == 0	Failure	Failure	Failed [217 / 300] Result: -.8660254040e-9-2.000000001I Test Values: {nu = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} Result: -.8660254040e-9-2.000000001I Test Values: {nu = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} ... skip entries to safe data	Failed [240 / 300] Result: Complex[1.1102230246251565^-16, 2.0] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.1102230246251565^-16, 2.0] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
10.2.E2	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{% k=0}^{\infty}(-1)^{k}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1% \right)}}}$ `\BesselJ{\nu}@{z} = (\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, z) = ((1)/(2)z)^(nu) sum((- 1)^(k)(((1)/(4)(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity)	BesselJ[\[Nu], z] == (Divide[1,2]z)^\[Nu] Sum[(- 1)^(k)Divide[(Divide[1,4](z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 70]
10.2.E3	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{J_{\nu}\left(z\right)% \cos\left(\nu\pi\right)-J_{-\nu}\left(z\right)}{\sin\left(\nu\pi\right)}}}$ `\BesselY{\nu}@{z} = \frac{\BesselJ{\nu}@{z}\cos@{\nu\pi}-\BesselJ{-\nu}@{z}}{\sin@{\nu\pi}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = (BesselJ(nu, z)cos(nuPi)- BesselJ(- nu, z))/(sin(nu*Pi))	BesselY[\[Nu], z] == Divide[BesselJ[\[Nu], z]Cos[\[Nu]Pi]- BesselJ[- \[Nu], z],Sin[\[Nu]*Pi]]	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4#Ex1	${\displaystyle{\displaystyle J_{-n}\left(z\right)=(-1)^{n}J_{n}\left(z\right)}}$ `\BesselJ{-n}@{z} = (-1)^{n}\BesselJ{n}@{z}`	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	BesselJ(- n, z) = (- 1)^(n)* BesselJ(n, z)	BesselJ[- n, z] == (- 1)^(n)* BesselJ[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex2	${\displaystyle{\displaystyle Y_{-n}\left(z\right)=(-1)^{n}Y_{n}\left(z\right)}}$ `\BesselY{-n}@{z} = (-1)^{n}\BesselY{n}@{z}`	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0,\Re((-(-n))+k+1)>0}}$	BesselY(- n, z) = (- 1)^(n)* BesselY(n, z)	BesselY[- n, z] == (- 1)^(n)* BesselY[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex3	${\displaystyle{\displaystyle{H^{(1)}_{-n}}\left(z\right)=(-1)^{n}{H^{(1)}_{n}}% \left(z\right)}}$ `\HankelH{1}{-n}@{z} = (-1)^{n}\HankelH{1}{n}@{z}`		HankelH1(- n, z) = (- 1)^(n)* HankelH1(n, z)	HankelH1[- n, z] == (- 1)^(n)* HankelH1[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex4	${\displaystyle{\displaystyle{H^{(2)}_{-n}}\left(z\right)=(-1)^{n}{H^{(2)}_{n}}% \left(z\right)}}$ `\HankelH{2}{-n}@{z} = (-1)^{n}\HankelH{2}{n}@{z}`		HankelH2(- n, z) = (- 1)^(n)* HankelH2(n, z)	HankelH2[- n, z] == (- 1)^(n)* HankelH2[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.4#Ex5	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=J_{\nu}\left(z\right% )+iY_{\nu}\left(z\right)}}$ `\HankelH{1}{\nu}@{z} = \BesselJ{\nu}@{z}+i\BesselY{\nu}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	HankelH1(nu, z) = BesselJ(nu, z)+ I*BesselY(nu, z)	HankelH1[\[Nu], z] == BesselJ[\[Nu], z]+ I*BesselY[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
10.4#Ex6	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=J_{\nu}\left(z\right% )-iY_{\nu}\left(z\right)}}$ `\HankelH{2}{\nu}@{z} = \BesselJ{\nu}@{z}-i\BesselY{\nu}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	HankelH2(nu, z) = BesselJ(nu, z)- I*BesselY(nu, z)	HankelH2[\[Nu], z] == BesselJ[\[Nu], z]- I*BesselY[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
10.4#Ex7	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2}\left({H^{(1)}_{% \nu}}\left(z\right)+{H^{(2)}_{\nu}}\left(z\right)\right)}}$ `\BesselJ{\nu}@{z} = \frac{1}{2}\left(\HankelH{1}{\nu}@{z}+\HankelH{2}{\nu}@{z}\right)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, z) = (1)/(2)*(HankelH1(nu, z)+ HankelH2(nu, z))	BesselJ[\[Nu], z] == Divide[1,2]*(HankelH1[\[Nu], z]+ HankelH2[\[Nu], z])	Successful	Successful	-	Successful [Tested: 70]
10.4#Ex8	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{2i}\left({H^{(1)}_% {\nu}}\left(z\right)-{H^{(2)}_{\nu}}\left(z\right)\right)}}$ `\BesselY{\nu}@{z} = \frac{1}{2i}\left(\HankelH{1}{\nu}@{z}-\HankelH{2}{\nu}@{z}\right)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = (1)/(2I)(HankelH1(nu, z)- HankelH2(nu, z))	BesselY[\[Nu], z] == Divide[1,2I](HankelH1[\[Nu], z]- HankelH2[\[Nu], z])	Successful	Successful	-	Successful [Tested: 70]
10.4.E5	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\csc\left(\nu\pi\right)% \left(Y_{-\nu}\left(z\right)-Y_{\nu}\left(z\right)\cos\left(\nu\pi\right)% \right)}}$ `\BesselJ{\nu}@{z} = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-\BesselY{\nu}@{z}\cos@{\nu\pi}\right)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-(-\nu))+k+1% )>0}}$	BesselJ(nu, z) = csc(nuPi)(BesselY(- nu, z)- BesselY(nu, z)cos(nuPi))	BesselJ[\[Nu], z] == Csc[\[Nu]Pi](BesselY[- \[Nu], z]- BesselY[\[Nu], z]Cos[\[Nu]Pi])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4#Ex9	${\displaystyle{\displaystyle{H^{(1)}_{-\nu}}\left(z\right)=e^{\nu\pi i}{H^{(1)% }_{\nu}}\left(z\right)}}$ `\HankelH{1}{-\nu}@{z} = e^{\nu\pi i}\HankelH{1}{\nu}@{z}`		HankelH1(- nu, z) = exp(nuPiI)*HankelH1(nu, z)	HankelH1[- \[Nu], z] == Exp[\[Nu]PiI]*HankelH1[\[Nu], z]	Successful	Failure	-	Successful [Tested: 70]
10.4#Ex10	${\displaystyle{\displaystyle{H^{(2)}_{-\nu}}\left(z\right)=e^{-\nu\pi i}{H^{(2% )}_{\nu}}\left(z\right)}}$ `\HankelH{2}{-\nu}@{z} = e^{-\nu\pi i}\HankelH{2}{\nu}@{z}`		HankelH2(- nu, z) = exp(- nuPiI)*HankelH2(nu, z)	HankelH2[- \[Nu], z] == Exp[- \[Nu]PiI]*HankelH2[\[Nu], z]	Successful	Failure	-	Successful [Tested: 70]
10.4.E7	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi% \right)\left(e^{-\nu\pi i}J_{\nu}\left(z\right)-J_{-\nu}\left(z\right)\right)}}$ `\HankelH{1}{\nu}@{z} = i\csc@{\nu\pi}\left(e^{-\nu\pi i}\BesselJ{\nu}@{z}-\BesselJ{-\nu}@{z}\right)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	HankelH1(nu, z) = Icsc(nuPi)(exp(- nuPiI)BesselJ(nu, z)- BesselJ(- nu, z))	HankelH1[\[Nu], z] == ICsc[\[Nu]Pi](Exp[- \[Nu]PiI]BesselJ[\[Nu], z]- BesselJ[- \[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4.E7	${\displaystyle{\displaystyle i\csc\left(\nu\pi\right)\left(e^{-\nu\pi i}J_{\nu% }\left(z\right)-J_{-\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(Y_{% -\nu}\left(z\right)-e^{-\nu\pi i}Y_{\nu}\left(z\right)\right)}}$ `i\csc@{\nu\pi}\left(e^{-\nu\pi i}\BesselJ{\nu}@{z}-\BesselJ{-\nu}@{z}\right) = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-e^{-\nu\pi i}\BesselY{\nu}@{z}\right)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-(-\nu))+k+1% )>0}}$	Icsc(nuPi)(exp(- nuPiI)BesselJ(nu, z)- BesselJ(- nu, z)) = csc(nuPi)(BesselY(- nu, z)- exp(- nuPiI)*BesselY(nu, z))	ICsc[\[Nu]Pi](Exp[- \[Nu]PiI]BesselJ[\[Nu], z]- BesselJ[- \[Nu], z]) == Csc[\[Nu]Pi](BesselY[- \[Nu], z]- Exp[- \[Nu]PiI]*BesselY[\[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4.E8	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi% \right)\left(J_{-\nu}\left(z\right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)}}$ `\HankelH{2}{\nu}@{z} = i\csc@{\nu\pi}\left(\BesselJ{-\nu}@{z}-e^{\nu\pi i}\BesselJ{\nu}@{z}\right)`	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}}$	HankelH2(nu, z) = Icsc(nuPi)(BesselJ(- nu, z)- exp(nuPiI)BesselJ(nu, z))	HankelH2[\[Nu], z] == ICsc[\[Nu]Pi](BesselJ[- \[Nu], z]- Exp[\[Nu]PiI]BesselJ[\[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.4.E8	${\displaystyle{\displaystyle i\csc\left(\nu\pi\right)\left(J_{-\nu}\left(z% \right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(% Y_{-\nu}\left(z\right)-e^{\nu\pi i}Y_{\nu}\left(z\right)\right)}}$ `i\csc@{\nu\pi}\left(\BesselJ{-\nu}@{z}-e^{\nu\pi i}\BesselJ{\nu}@{z}\right) = \csc@{\nu\pi}\left(\BesselY{-\nu}@{z}-e^{\nu\pi i}\BesselY{\nu}@{z}\right)`	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0,\Re((-(-\nu))+k+1% )>0}}$	Icsc(nuPi)(BesselJ(- nu, z)- exp(nuPiI)BesselJ(nu, z)) = csc(nuPi)(BesselY(- nu, z)- exp(nuPiI)*BesselY(nu, z))	ICsc[\[Nu]Pi](BesselJ[- \[Nu], z]- Exp[\[Nu]PiI]BesselJ[\[Nu], z]) == Csc[\[Nu]Pi](BesselY[- \[Nu], z]- Exp[\[Nu]PiI]*BesselY[\[Nu], z])	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.5.E1	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),J_{-\nu}% \left(z\right)\right\}=J_{\nu+1}\left(z\right)J_{-\nu}\left(z\right)+J_{\nu}% \left(z\right)J_{-\nu-1}\left(z\right)}}$ `\Wronskian@{\BesselJ{\nu}@{z},\BesselJ{-\nu}@{z}} = \BesselJ{\nu+1}@{z}\BesselJ{-\nu}@{z}+\BesselJ{\nu}@{z}\BesselJ{-\nu-1}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((\nu+1)+k+1)>% 0,\Re((-\nu-1)+k+1)>0}}$	(BesselJ(nu, z))diff(BesselJ(- nu, z), z)-diff(BesselJ(nu, z), z)(BesselJ(- nu, z)) = BesselJ(nu + 1, z)BesselJ(- nu, z)+ BesselJ(nu, z)BesselJ(- nu - 1, z)	Wronskian[{BesselJ[\[Nu], z], BesselJ[- \[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]BesselJ[- \[Nu], z]+ BesselJ[\[Nu], z]BesselJ[- \[Nu]- 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E1	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right)J_{-\nu}\left(z\right)+J_{% \nu}\left(z\right)J_{-\nu-1}\left(z\right)=-2\sin\left(\nu\pi\right)/(\pi z)}}$ `\BesselJ{\nu+1}@{z}\BesselJ{-\nu}@{z}+\BesselJ{\nu}@{z}\BesselJ{-\nu-1}@{z} = -2\sin@{\nu\pi}/(\pi z)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((\nu+1)+k+1)>% 0,\Re((-\nu-1)+k+1)>0}}$	BesselJ(nu + 1, z)BesselJ(- nu, z)+ BesselJ(nu, z)BesselJ(- nu - 1, z) = - 2sin(nuPi)/(Pi*z)	BesselJ[\[Nu]+ 1, z]BesselJ[- \[Nu], z]+ BesselJ[\[Nu], z]BesselJ[- \[Nu]- 1, z] == - 2Sin[\[Nu]Pi]/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E2	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),Y_{\nu}% \left(z\right)\right\}=J_{\nu+1}\left(z\right)Y_{\nu}\left(z\right)-J_{\nu}% \left(z\right)Y_{\nu+1}\left(z\right)}}$ `\Wronskian@{\BesselJ{\nu}@{z},\BesselY{\nu}@{z}} = \BesselJ{\nu+1}@{z}\BesselY{\nu}@{z}-\BesselJ{\nu}@{z}\BesselY{\nu+1}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0,\Re((-\nu)+k+1)>% 0,\Re((-(\nu+1))+k+1)>0}}$	(BesselJ(nu, z))diff(BesselY(nu, z), z)-diff(BesselJ(nu, z), z)(BesselY(nu, z)) = BesselJ(nu + 1, z)BesselY(nu, z)- BesselJ(nu, z)BesselY(nu + 1, z)	Wronskian[{BesselJ[\[Nu], z], BesselY[\[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]BesselY[\[Nu], z]- BesselJ[\[Nu], z]BesselY[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E2	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right)Y_{\nu}\left(z\right)-J_{% \nu}\left(z\right)Y_{\nu+1}\left(z\right)=2/(\pi z)}}$ `\BesselJ{\nu+1}@{z}\BesselY{\nu}@{z}-\BesselJ{\nu}@{z}\BesselY{\nu+1}@{z} = 2/(\pi z)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0,\Re((-\nu)+k+1)>% 0,\Re((-(\nu+1))+k+1)>0}}$	BesselJ(nu + 1, z)BesselY(nu, z)- BesselJ(nu, z)BesselY(nu + 1, z) = 2/(Pi*z)	BesselJ[\[Nu]+ 1, z]BesselY[\[Nu], z]- BesselJ[\[Nu], z]BesselY[\[Nu]+ 1, z] == 2/(Pi*z)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E3	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{% \nu}}\left(z\right)\right\}=J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z% \right)-J_{\nu}\left(z\right){H^{(1)}_{\nu+1}}\left(z\right)}}$ `\Wronskian@{\BesselJ{\nu}@{z},\HankelH{1}{\nu}@{z}} = \BesselJ{\nu+1}@{z}\HankelH{1}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{1}{\nu+1}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	(BesselJ(nu, z))diff(HankelH1(nu, z), z)-diff(BesselJ(nu, z), z)(HankelH1(nu, z)) = BesselJ(nu + 1, z)HankelH1(nu, z)- BesselJ(nu, z)HankelH1(nu + 1, z)	Wronskian[{BesselJ[\[Nu], z], HankelH1[\[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]HankelH1[\[Nu], z]- BesselJ[\[Nu], z]HankelH1[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E3	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z% \right)-J_{\nu}\left(z\right){H^{(1)}_{\nu+1}}\left(z\right)=2i/(\pi z)}}$ `\BesselJ{\nu+1}@{z}\HankelH{1}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{1}{\nu+1}@{z} = 2i/(\pi z)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	BesselJ(nu + 1, z)HankelH1(nu, z)- BesselJ(nu, z)HankelH1(nu + 1, z) = 2I/(Piz)	BesselJ[\[Nu]+ 1, z]HankelH1[\[Nu], z]- BesselJ[\[Nu], z]HankelH1[\[Nu]+ 1, z] == 2I/(Piz)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(2)}_{% \nu}}\left(z\right)\right\}=J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z% \right)-J_{\nu}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)}}$ `\Wronskian@{\BesselJ{\nu}@{z},\HankelH{2}{\nu}@{z}} = \BesselJ{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{2}{\nu+1}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	(BesselJ(nu, z))diff(HankelH2(nu, z), z)-diff(BesselJ(nu, z), z)(HankelH2(nu, z)) = BesselJ(nu + 1, z)HankelH2(nu, z)- BesselJ(nu, z)HankelH2(nu + 1, z)	Wronskian[{BesselJ[\[Nu], z], HankelH2[\[Nu], z]}, z] == BesselJ[\[Nu]+ 1, z]HankelH2[\[Nu], z]- BesselJ[\[Nu], z]HankelH2[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E4	${\displaystyle{\displaystyle J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z% \right)-J_{\nu}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-2i/(\pi z)}}$ `\BesselJ{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\BesselJ{\nu}@{z}\HankelH{2}{\nu+1}@{z} = -2i/(\pi z)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	BesselJ(nu + 1, z)HankelH2(nu, z)- BesselJ(nu, z)HankelH2(nu + 1, z) = - 2I/(Piz)	BesselJ[\[Nu]+ 1, z]HankelH2[\[Nu], z]- BesselJ[\[Nu], z]HankelH2[\[Nu]+ 1, z] == - 2I/(Piz)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.5.E5	${\displaystyle{\displaystyle\mathscr{W}\left\{{H^{(1)}_{\nu}}\left(z\right),{H% ^{(2)}_{\nu}}\left(z\right)\right\}={H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{% \nu}}\left(z\right)-{H^{(1)}_{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z% \right)}}$ `\Wronskian@{\HankelH{1}{\nu}@{z},\HankelH{2}{\nu}@{z}} = \HankelH{1}{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\HankelH{1}{\nu}@{z}\HankelH{2}{\nu+1}@{z}`		(HankelH1(nu, z))diff(HankelH2(nu, z), z)-diff(HankelH1(nu, z), z)(HankelH2(nu, z)) = HankelH1(nu + 1, z)HankelH2(nu, z)- HankelH1(nu, z)HankelH2(nu + 1, z)	Wronskian[{HankelH1[\[Nu], z], HankelH2[\[Nu], z]}, z] == HankelH1[\[Nu]+ 1, z]HankelH2[\[Nu], z]- HankelH1[\[Nu], z]HankelH2[\[Nu]+ 1, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 70]
10.5.E5	${\displaystyle{\displaystyle{H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}% \left(z\right)-{H^{(1)}_{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-4% i/(\pi z)}}$ `\HankelH{1}{\nu+1}@{z}\HankelH{2}{\nu}@{z}-\HankelH{1}{\nu}@{z}\HankelH{2}{\nu+1}@{z} = -4i/(\pi z)`		HankelH1(nu + 1, z)HankelH2(nu, z)- HankelH1(nu, z)HankelH2(nu + 1, z) = - 4I/(Piz)	HankelH1[\[Nu]+ 1, z]HankelH2[\[Nu], z]- HankelH1[\[Nu], z]HankelH2[\[Nu]+ 1, z] == - 4I/(Piz)	Failure	Successful	Successful [Tested: 70]	Successful [Tested: 70]
10.6#E3X	${\displaystyle{\displaystyle\displaystyle J_{0}'\left(z\right)=-J_{1}\left(z% \right)}}$ `\displaystyle\BesselJ{0}'@{z} = -\BesselJ{1}@{z}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0}}$	diff( BesselJ(0, z), z$(1) ) = - BesselJ(1, z)	D[BesselJ[0, z], {z, 1}] == - BesselJ[1, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#E3X	${\displaystyle{\displaystyle\displaystyle Y_{0}'\left(z\right)=-Y_{1}\left(z% \right)}}$ `\displaystyle\BesselY{0}'@{z} = -\BesselY{1}@{z}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0,\Re((-0)+k+1)>0,\Re((-1)% +k+1)>0}}$	diff( BesselY(0, z), z$(1) ) = - BesselY(1, z)	D[BesselY[0, z], {z, 1}] == - BesselY[1, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#E3Xa	${\displaystyle{\displaystyle\displaystyle{H^{(1)}_{0}}'\left(z\right)=-{H^{(1)% }_{1}}\left(z\right)}}$ `\displaystyle\HankelH{1}{0}'@{z} = -\HankelH{1}{1}@{z}`		diff( HankelH1(0, z), z$(1) ) = - HankelH1(1, z)	D[HankelH1[0, z], {z, 1}] == - HankelH1[1, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#E3Xa	${\displaystyle{\displaystyle\displaystyle{H^{(2)}_{0}}'\left(z\right)=-{H^{(2)% }_{1}}\left(z\right)}}$ `\displaystyle\HankelH{2}{0}'@{z} = -\HankelH{2}{1}@{z}`		diff( HankelH2(0, z), z$(1) ) = - HankelH2(1, z)	D[HankelH2[0, z], {z, 1}] == - HankelH2[1, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#Ex5	${\displaystyle{\displaystyle f_{\nu-1}(z)+f_{\nu+1}(z)=(2\nu/\lambda)z^{-q}f_{% \nu}(z)}}$ `f_{\nu-1}(z)+f_{\nu+1}(z) = (2\nu/\lambda)z^{-q}f_{\nu}(z)`		f[nu - 1](z)+ f[nu + 1](z) = (2nu/lambda)(z)^(- q)* f[nu](z)	Subscript[f, \[Nu]- 1][z]+ Subscript[f, \[Nu]+ 1][z] == (2\[Nu]/\[Lambda])(z)^(- q)* Subscript[f, \[Nu]][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#Ex15	${\displaystyle{\displaystyle p_{\nu+1}-p_{\nu-1}=-\frac{2\nu}{a}q_{\nu}-\frac{% 2\nu}{b}r_{\nu}}}$ `p_{\nu+1}-p_{\nu-1} = -\frac{2\nu}{a}q_{\nu}-\frac{2\nu}{b}r_{\nu}`		p[nu + 1]- p[nu - 1] = -(2nu)/(a)q[nu]-(2nu)/(b)r[nu]	Subscript[p, \[Nu]+ 1]- Subscript[p, \[Nu]- 1] == -Divide[2\[Nu],a]Subscript[q, \[Nu]]-Divide[2\[Nu],b]Subscript[r, \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#Ex16	${\displaystyle{\displaystyle q_{\nu+1}+r_{\nu}=\frac{\nu}{a}p_{\nu}-\frac{\nu+% 1}{b}p_{\nu+1}}}$ `q_{\nu+1}+r_{\nu} = \frac{\nu}{a}p_{\nu}-\frac{\nu+1}{b}p_{\nu+1}`		q[nu + 1]+ r[nu] = (nu)/(a)p[nu]-(nu + 1)/(b)p[nu + 1]	Subscript[q, \[Nu]+ 1]+ Subscript[r, \[Nu]] == Divide[\[Nu],a]Subscript[p, \[Nu]]-Divide[\[Nu]+ 1,b]Subscript[p, \[Nu]+ 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#Ex17	${\displaystyle{\displaystyle r_{\nu+1}+q_{\nu}=\frac{\nu}{b}p_{\nu}-\frac{\nu+% 1}{a}p_{\nu+1}}}$ `r_{\nu+1}+q_{\nu} = \frac{\nu}{b}p_{\nu}-\frac{\nu+1}{a}p_{\nu+1}`		r[nu + 1]+ q[nu] = (nu)/(b)p[nu]-(nu + 1)/(a)p[nu + 1]	Subscript[r, \[Nu]+ 1]+ Subscript[q, \[Nu]] == Divide[\[Nu],b]Subscript[p, \[Nu]]-Divide[\[Nu]+ 1,a]Subscript[p, \[Nu]+ 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6#Ex18	${\displaystyle{\displaystyle s_{\nu}=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-% 1}-\frac{\nu^{2}}{ab}p_{\nu}}}$ `s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-\frac{\nu^{2}}{ab}p_{\nu}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	(diff( BesselJ(nu, a), a$(1) )diff( BesselY(nu, b), b$(1) )- diff( BesselJ(nu, b), b$(1) )diff( BesselY(nu, a), a$(1) )) = (1)/(2)p[nu + 1]+(1)/(2)p[nu - 1]-((nu)^(2))/(ab)p[nu]	(D[BesselJ[\[Nu], a], {a, 1}]D[BesselY[\[Nu], b], {b, 1}]- D[BesselJ[\[Nu], b], {b, 1}]D[BesselY[\[Nu], a], {a, 1}]) == Divide[1,2]Subscript[p, \[Nu]+ 1]+Divide[1,2]Subscript[p, \[Nu]- 1]-Divide[\[Nu]^(2),ab]Subscript[p, \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.6.E10	${\displaystyle{\displaystyle p_{\nu}s_{\nu}-q_{\nu}r_{\nu}=4/(\pi^{2}ab)}}$ `p_{\nu}s_{\nu}-q_{\nu}r_{\nu} = 4/(\pi^{2}ab)`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	p[nu](diff( BesselJ(nu, a), a$(1) )diff( BesselY(nu, b), b$(1) )- diff( BesselJ(nu, b), b$(1) )diff( BesselY(nu, a), a$(1) ))- q[nu]r[nu] = 4/((Pi)^(2)* a*b)	Subscript[p, \[Nu]](D[BesselJ[\[Nu], a], {a, 1}]D[BesselY[\[Nu], b], {b, 1}]- D[BesselJ[\[Nu], b], {b, 1}]D[BesselY[\[Nu], a], {a, 1}])- Subscript[q, \[Nu]]Subscript[r, \[Nu]] == 4/((Pi)^(2)* a*b)	Skipped - no semantic math	Skipped - no semantic math	-	-
10.8.E1	${\displaystyle{\displaystyle Y_{n}\left(z\right)=-\frac{(\tfrac{1}{2}z)^{-n}}{% \pi}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+% \frac{2}{\pi}\ln\left(\tfrac{1}{2}z\right)J_{n}\left(z\right)-\frac{(\tfrac{1}% {2}z)^{n}}{\pi}\sum_{k=0}^{\infty}(\psi\left(k+1\right)+\psi\left(n+k+1\right)% )\frac{(-\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!}}}$ `\BesselY{n}@{z} = -\frac{(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+\frac{2}{\pi}\ln@{\tfrac{1}{2}z}\BesselJ{n}@{z}-\frac{(\tfrac{1}{2}z)^{n}}{\pi}\sum_{k=0}^{\infty}(\digamma@{k+1}+\digamma@{n+k+1})\frac{(-\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!}`	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re((-n)+k+1)>0}}$	BesselY(n, z) = -(((1)/(2)z)^(- n))/(Pi)sum((factorial(n - k - 1))/(factorial(k))((1)/(4)(z)^(2))^(k), k = 0..n - 1)+(2)/(Pi)ln((1)/(2)z)BesselJ(n, z)-(((1)/(2)z)^(n))/(Pi)sum((Psi(k + 1)+ Psi(n + k + 1))((-(1)/(4)(z)^(2))^(k))/(factorial(k)factorial(n + k)), k = 0..infinity)	BesselY[n, z] == -Divide[(Divide[1,2]z)^(- n),Pi]Sum[Divide[(n - k - 1)!,(k)!](Divide[1,4](z)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]+Divide[2,Pi]Log[Divide[1,2]z]BesselJ[n, z]-Divide[(Divide[1,2]z)^(n),Pi]Sum[(PolyGamma[k + 1]+ PolyGamma[n + k + 1])Divide[(-Divide[1,4](z)^(2))^(k),(k)!(n + k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [6 / 21] Result: Plus[-0.4244131815783875, Times[0.4244131815783876, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-4, []], Times[Plus[12, Times[8, ]], [Plus[1, ]]], Times[Plus[-16, Times[-16, ], Times[-4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[2, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], 1], Equal[[2], Plus[1, Times[4, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 2], Times[16, Power[1.5, -4], Plus[2, Times[Rational[1, 4], Power[1.5, 2]]]]]], Equal[[4], Times[Rational[32, 3], Power[1.5, -6], Plus[3, Times[Rational[1, 4], Power[1.5, 2]]], Plus[12, Times[Rational[1, 16], Power[1.5, 4]]]]]}]][1.0]]], {Rule[n, 1], Rule[z, 1.5]} Result: Plus[-0.8841941282883073, Times[0.3183098861837907, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-4, []], Times[Plus[12, Times[8, ]], [Plus[1, ]]], Times[Plus[-16, Times[-16, ], Times[-4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[2, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], 1], Equal[[2], Plus[1, Times[4, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 2], Times[16, Power[1.5, -4], Plus[2, Times[Rational[1, 4], Power[1.5, 2]]]]]], Equal[[4], Times[Rational[32, 3], Power[1.5, -6], Plus[3, Times[Rational[1, 4], Power[1.5, 2]]], Plus[12, Times[Rational[1, 16], Power[1.5, 4]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, 1.5]} ... skip entries to safe data
10.8.E2	${\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(% \tfrac{1}{2}z\right)+\gamma\right)J_{0}\left(z\right)+\frac{2}{\pi}\left(\frac% {\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(% 2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}% -\cdots\right)}}$ `\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\dotsi\right)`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0}}$	BesselY(0, z) = (2)/(Pi)(ln((1)/(2)z)+ gamma)BesselJ(0, z)+(2)/(Pi)(((1)/(4)(z)^(2))/((factorial(1))^(2))-(1 +(1)/(2))(((1)/(4)(z)^(2))^(2))/((factorial(2))^(2))+(1 +(1)/(2)+(1)/(3))(((1)/(4)*(z)^(2))^(3))/((factorial(3))^(2))- ..)	BesselY[0, z] == Divide[2,Pi](Log[Divide[1,2]z]+ EulerGamma)BesselJ[0, z]+Divide[2,Pi](Divide[Divide[1,4](z)^(2),((1)!)^(2)]-(1 +Divide[1,2])Divide[(Divide[1,4](z)^(2))^(2),((2)!)^(2)]+(1 +Divide[1,2]+Divide[1,3])Divide[(Divide[1,4]*(z)^(2))^(3),((3)!)^(2)]- \[Ellipsis])	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[0.08653583575184755, 0.12491815695491987], Times[-0.6366197723675814, Plus[Complex[0.13592303240740744, 0.19620888054491187], Times[-1.0, …]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.07160606681826986, -0.15074612001799426], Times[-0.6366197723675814, Plus[Complex[-0.11248553240740736, -0.23680382134730746], Times[-1.0, …]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.8.E3	${\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\mu}\left(z\right)=(% \tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(-\tfrac{1}{% 4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}}}$ `\BesselJ{\nu}@{z}\BesselJ{\mu}@{z} = (\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(-\tfrac{1}{4}z^{2})^{k}}{k!\EulerGamma@{\nu+k+1}\EulerGamma@{\mu+k+1}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\mu)+k+1)>0,\Re(\mu+k+1)>0}}$	BesselJ(nu, z)BesselJ(mu, z) = ((1)/(2)z)^(nu + mu)* sum((nu + mu + k + 1[k](-(1)/(4)(z)^(2))^(k))/(factorial(k)GAMMA(nu + k + 1)GAMMA(mu + k + 1)), k = 0..infinity)	BesselJ[\[Nu], z]BesselJ[\[Mu], z] == (Divide[1,2]z)^(\[Nu]+ \[Mu])* Sum[Divide[Subscript[\[Nu]+ \[Mu]+ k + 1, k](-Divide[1,4](z)^(2))^(k),(k)!Gamma[\[Nu]+ k + 1]Gamma[\[Mu]+ k + 1]], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [300 / 300] Result: Plus[Complex[0.18482793500467376, -0.06270111308873656], Times[Complex[-0.17426361621858172, -0.037827155645948574], NSum[Times[Power[Times[Rational[-1, 4], Power[E, Times[Complex[0, Rational[1, 3]], Pi]]], k], Power[Factorial[k], -1], Power[Gamma[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]], -2], Subscript[Plus[1, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], k]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.47215054540190965, -0.036453907426047115], Times[Complex[-0.27630938504679325, 0.26010894184513544], NSum[Times[Power[Times[Rational[-1, 4], Power[E, Times[Complex[0, Rational[1, 3]], Pi]]], k], Power[Factorial[k], -1], Power[Gamma[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]], -1], Power[Gamma[Plus[1, Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k]], -1], Subscript[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], k]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.9.E1	${\displaystyle{\displaystyle J_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta\right)\mathrm{d}\theta}}$ `\BesselJ{0}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta}`	${\displaystyle{\displaystyle\Re(0+k+1)>0}}$	BesselJ(0, z) = (1)/(Pi)int(cos(zsin(theta)), theta = 0..Pi)	BesselJ[0, z] == Divide[1,Pi]Integrate[Cos[zSin[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]	Successful	Successful	-	Failed [4 / 7] Result: Complex[0.1024204169391214, -0.20298051839359257] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.35155242920280916, 0.2300320660405755] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.9.E1	${\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta% \right)\mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\cos\theta\right% )\mathrm{d}\theta}}$ `\frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}\diff{\theta}`	${\displaystyle{\displaystyle\Re(0+k+1)>0}}$	(1)/(Pi)int(cos(zsin(theta)), theta = 0..Pi) = (1)/(Pi)int(cos(zcos(theta)), theta = 0..Pi)	Divide[1,Pi]Integrate[Cos[zSin[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[1,Pi]Integrate[Cos[zCos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
10.9.E2	${\displaystyle{\displaystyle J_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta-n\theta\right)\mathrm{d}\theta}}$ `\BesselJ{n}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	BesselJ(n, z) = (1)/(Pi)int(cos(zsin(theta)- n*theta), theta = 0..Pi)	BesselJ[n, z] == Divide[1,Pi]Integrate[Cos[zSin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 7]	Successful [Tested: 7]
10.9.E2	${\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-n% \theta\right)\mathrm{d}\theta=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos\theta}% \cos\left(n\theta\right)\mathrm{d}\theta}}$ `\frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta} = \frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\cos@{n\theta}\diff{\theta}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	(1)/(Pi)int(cos(zsin(theta)- ntheta), theta = 0..Pi) = ((I)^(- n))/(Pi)int(exp(Izcos(theta))cos(ntheta), theta = 0..Pi)	Divide[1,Pi]Integrate[Cos[zSin[\[Theta]]- n\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[(I)^(- n),Pi]Integrate[Exp[IzCos[\[Theta]]]Cos[n\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 7]	Skipped - Because timed out
10.9.E3	${\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{4}{\pi^{2}}\int_{0}^{% \frac{1}{2}\pi}\cos\left(z\cos\theta\right)\left(\gamma+\ln\left(2z{\sin^{2}}% \theta\right)\right)\mathrm{d}\theta}}$ `\BesselY{0}@{z} = \frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos@{z\cos@@{\theta}}\left(\EulerConstant+\ln@{2z\sin^{2}@@{\theta}}\right)\diff{\theta}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0}}$	BesselY(0, z) = (4)/((Pi)^(2))int(cos(zcos(theta))(gamma + ln(2z(sin(theta))^(2))), theta = 0..(1)/(2)Pi)	BesselY[0, z] == Divide[4,(Pi)^(2)]Integrate[Cos[zCos[\[Theta]]](EulerGamma + Log[2z(Sin[\[Theta]])^(2)]), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None]	Aborted	Aborted	Successful [Tested: 7]	Skipped - Because timed out
10.9.E4	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left% (z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta}}$ `\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta}`	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu+\tfrac% {1}{2})>0}}$	BesselJ(nu, z) = (((1)/(2)z)^(nu))/((Pi)^((1)/(2)) GAMMA(nu +(1)/(2)))int(cos(zcos(theta))(sin(theta))^(2nu), theta = 0..Pi)	BesselJ[\[Nu], z] == Divide[(Divide[1,2]z)^\[Nu],(Pi)^(Divide[1,2]) Gamma[\[Nu]+Divide[1,2]]]Integrate[Cos[zCos[\[Theta]]](Sin[\[Theta]])^(2\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None]	Error	Successful	-	Failed [20 / 35] Result: Complex[0.009683985979314524, -0.05759180507972181] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.21993206762171735, 0.08917811286212163] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
10.9.E4	${\displaystyle{\displaystyle\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}% \Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left(z\cos\theta\right)% (\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1% }{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2% }}\cos\left(zt\right)\mathrm{d}t}}$ `\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\cos@{zt}\diff{t}`	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu+\tfrac% {1}{2})>0}}$	(((1)/(2)z)^(nu))/((Pi)^((1)/(2)) GAMMA(nu +(1)/(2)))int(cos(zcos(theta))(sin(theta))^(2nu), theta = 0..Pi) = (2((1)/(2)z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))int((1 - (t)^(2))^(nu -(1)/(2)) cos(z*t), t = 0..1)	Divide[(Divide[1,2]z)^\[Nu],(Pi)^(Divide[1,2]) Gamma[\[Nu]+Divide[1,2]]]Integrate[Cos[zCos[\[Theta]]](Sin[\[Theta]])^(2\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[2(Divide[1,2]z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2]) Cos[z*t], {t, 0, 1}, GenerateConditions->None]	Error	Successful	-	Successful [Tested: 35]
10.9.E5	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu% }}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t% ^{2})^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t-\int_{0}^{\infty}e^{-zt}% (1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t\right)}}$ `\BesselY{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin@{zt}\diff{t}-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}\right)`	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\|\operatorname{ph}z\|<\tfrac{1% }{2}\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(\nu+\tfrac{1}{2})>0}}$	BesselY(nu, z) = (2((1)/(2)z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))(int((1 - (t)^(2))^(nu -(1)/(2)) sin(zt), t = 0..1)- int(exp(- zt)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity))	BesselY[\[Nu], z] == Divide[2(Divide[1,2]z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]](Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2]) Sin[zt], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- zt]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}, GenerateConditions->None])	Successful	Successful	-	Failed [15 / 25] Result: Complex[-0.9495382353861556, 0.46093572348323536] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]} Result: Complex[-0.7706973036767981, 0.20650772012904162] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]} ... skip entries to safe data
10.9.E6	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi% \right)}{\pi}\int_{0}^{\infty}e^{-z\sinh t-\nu t}\mathrm{d}t}}$ `\BesselJ{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{\sin@{\nu\pi}}{\pi}\int_{0}^{\infty}e^{-z\sinh@@{t}-\nu t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0}}$	BesselJ(nu, z) = (1)/(Pi)int(cos(zsin(theta)- nutheta), theta = 0..Pi)-(sin(nuPi))/(Pi)int(exp(- zsinh(t)- nu*t), t = 0..infinity)	BesselJ[\[Nu], z] == Divide[1,Pi]Integrate[Cos[zSin[\[Theta]]- \[Nu]\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[Sin[\[Nu]Pi],Pi]Integrate[Exp[- zSinh[t]- \[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [1 / 50] Result: -.1812319652 Test Values: {nu = -1/2, z = 3/2}	Skipped - Because timed out
10.9.E7	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{% \infty}\left(e^{\nu t}+e^{-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}% \mathrm{d}t}}$ `\BesselY{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\sin@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t}+e^{-\nu t}\cos@{\nu\pi}\right)e^{-z\sinh@@{t}}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = (1)/(Pi)int(sin(zsin(theta)- nutheta), theta = 0..Pi)-(1)/(Pi)int((exp(nut)+ exp(- nut)cos(nuPi))exp(- zsinh(t)), t = 0..infinity)	BesselY[\[Nu], z] == Divide[1,Pi]Integrate[Sin[zSin[\[Theta]]- \[Nu]\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[1,Pi]Integrate[(Exp[\[Nu]t]+ Exp[- \[Nu]t]Cos[\[Nu]Pi])Exp[- zSinh[t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9#Ex1	${\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2}{\pi}\int_{0}^{% \infty}\sin\left(x\cosh t-\tfrac{1}{2}\nu\pi\right)\cosh\left(\nu t\right)% \mathrm{d}t}}$ `\BesselJ{\nu}@{x} = \frac{2}{\pi}\int_{0}^{\infty}\sin@{x\cosh@@{t}-\tfrac{1}{2}\nu\pi}\cosh@{\nu t}\diff{t}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, x) = (2)/(Pi)int(sin(xcosh(t)-(1)/(2)nuPi)cosh(nut), t = 0..infinity)	BesselJ[\[Nu], x] == Divide[2,Pi]Integrate[Sin[xCosh[t]-Divide[1,2]\[Nu]Pi]Cosh[\[Nu]t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9#Ex2	${\displaystyle{\displaystyle Y_{\nu}\left(x\right)=-\frac{2}{\pi}\int_{0}^{% \infty}\cos\left(x\cosh t-\tfrac{1}{2}\nu\pi\right)\cosh\left(\nu t\right)% \mathrm{d}t}}$ `\BesselY{\nu}@{x} = -\frac{2}{\pi}\int_{0}^{\infty}\cos@{x\cosh@@{t}-\tfrac{1}{2}\nu\pi}\cosh@{\nu t}\diff{t}`	${\displaystyle{\displaystyle\|\Re\nu\|<1,x>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, x) = -(2)/(Pi)int(cos(xcosh(t)-(1)/(2)nuPi)cosh(nut), t = 0..infinity)	BesselY[\[Nu], x] == -Divide[2,Pi]Integrate[Cos[xCosh[t]-Divide[1,2]\[Nu]Pi]Cosh[\[Nu]t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9#Ex3	${\displaystyle{\displaystyle J_{0}\left(x\right)=\frac{2}{\pi}\int_{0}^{\infty% }\sin\left(x\cosh t\right)\mathrm{d}t}}$ `\BesselJ{0}@{x} = \frac{2}{\pi}\int_{0}^{\infty}\sin@{x\cosh@@{t}}\diff{t}`	${\displaystyle{\displaystyle x>0,\Re(0+k+1)>0}}$	BesselJ(0, x) = (2)/(Pi)int(sin(xcosh(t)), t = 0..infinity)	BesselJ[0, x] == Divide[2,Pi]Integrate[Sin[xCosh[t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9#Ex4	${\displaystyle{\displaystyle Y_{0}\left(x\right)=-\frac{2}{\pi}\int_{0}^{% \infty}\cos\left(x\cosh t\right)\mathrm{d}t}}$ `\BesselY{0}@{x} = -\frac{2}{\pi}\int_{0}^{\infty}\cos@{x\cosh@@{t}}\diff{t}`	${\displaystyle{\displaystyle x>0,\Re(0+k+1)>0,\Re((-0)+k+1)>0}}$	BesselY(0, x) = -(2)/(Pi)int(cos(xcosh(t)), t = 0..infinity)	BesselY[0, x] == -Divide[2,Pi]Integrate[Cos[xCosh[t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9.E10	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{e^{-\frac{1}{2% }\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t-\nu t}\mathrm{d}t}}$ `\HankelH{1}{\nu}@{z} = \frac{e^{-\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{iz\cosh@@{t}-\nu t}\diff{t}`	${\displaystyle{\displaystyle 0<\operatorname{ph}z,\operatorname{ph}z<\pi}}$	HankelH1(nu, z) = (exp(-(1)/(2)nuPiI))/(PiI)int(exp(Izcosh(t)- nut), t = - infinity..infinity)	HankelH1[\[Nu], z] == Divide[Exp[-Divide[1,2]\[Nu]PiI],PiI]Integrate[Exp[IzCosh[t]- \[Nu]t], {t, - Infinity, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E11	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2% }\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-\nu t}\mathrm{d}t}}$ `\HankelH{2}{\nu}@{z} = -\frac{e^{\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh@@{t}-\nu t}\diff{t}`	${\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<0}}$	HankelH2(nu, z) = -(exp((1)/(2)nuPiI))/(PiI)int(exp(- Izcosh(t)- nut), t = - infinity..infinity)	HankelH2[\[Nu], z] == -Divide[Exp[Divide[1,2]\[Nu]PiI],PiI]Integrate[Exp[- IzCosh[t]- \[Nu]t], {t, - Infinity, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9#Ex5	${\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2(\tfrac{1}{2}x)^{-% \nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}% \frac{\sin\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}}}}$ `\BesselJ{\nu}@{x} = \frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\tfrac{1}{2}-\nu}}\int_{1}^{\infty}\frac{\sin@{xt}\diff{t}}{(t^{2}-1)^{\nu+\frac{1}{2}}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}-\nu)>0}}$	BesselJ(nu, x) = (2((1)/(2)x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))int((sin(xt))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity)	BesselJ[\[Nu], x] == Divide[2(Divide[1,2]x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]Integrate[Divide[Sin[xt],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}, GenerateConditions->None]	Successful	Aborted	-	Successful [Tested: 15]
10.9#Ex6	${\displaystyle{\displaystyle Y_{\nu}\left(x\right)=-\frac{2(\tfrac{1}{2}x)^{-% \nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}% \frac{\cos\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}}}}$ `\BesselY{\nu}@{x} = -\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\tfrac{1}{2}-\nu}}\int_{1}^{\infty}\frac{\cos@{xt}\diff{t}}{(t^{2}-1)^{\nu+\frac{1}{2}}}`	${\displaystyle{\displaystyle\|\Re\nu\|<\tfrac{1}{2},x>0,\Re(\tfrac{1}{2}-\nu)>0,% \Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, x) = -(2((1)/(2)x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))int((cos(xt))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity)	BesselY[\[Nu], x] == -Divide[2(Divide[1,2]x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]Integrate[Divide[Cos[xt],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}, GenerateConditions->None]	Successful	Aborted	-	Skip - No test values generated
10.9.E13	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}J_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t% -\nu t}\mathrm{d}t}}$ `\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\BesselJ{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = \frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos@@{\theta}}\cos@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{\sin@{\nu\pi}}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh@@{t}-z\sinh@@{t}-\nu t}\diff{t}`	${\displaystyle{\displaystyle\Re\left(z+\zeta\right)>0,\Re(\nu+k+1)>0}}$	((z + zeta)/(z - zeta))^((1)/(2)nu) BesselJ(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = (1)/(Pi)int(exp(zetacos(theta))cos(zsin(theta)- nutheta), theta = 0..Pi)-(sin(nuPi))/(Pi)int(exp(- zetacosh(t)- zsinh(t)- nut), t = 0..infinity)	(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]\[Nu]) BesselJ[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == Divide[1,Pi]Integrate[Exp[\[Zeta]Cos[\[Theta]]]Cos[zSin[\[Theta]]- \[Nu]\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[Sin[\[Nu]Pi],Pi]Integrate[Exp[- \[Zeta]Cosh[t]- zSinh[t]- \[Nu]t], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E14	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}Y_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t+\zeta\cosh t}+e^{-\nu t-\zeta% \cosh t}\cos\left(\nu\pi\right)\right)\e^{-z\sinh t}\mathrm{d}t}}$ `\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\BesselY{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = \frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos@@{\theta}}\sin@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t+\zeta\cosh@@{t}}+e^{-\nu t-\zeta\cosh@@{t}}\cos@{\nu\pi}\right)\e^{-z\sinh@@{t}}\diff{t}`	${\displaystyle{\displaystyle\Re\left(z+\zeta\right)>0,\Re\left(z-\zeta\right)>% 0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	((z + zeta)/(z - zeta))^((1)/(2)nu) BesselY(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = (1)/(Pi)int(exp(zetacos(theta))sin(zsin(theta)- nutheta), theta = 0..Pi)-(1)/(Pi)int((exp(nut + zetacosh(t))+ exp(- nut - zetacosh(t))cos(nuPi))* exp(- z*sinh(t)), t = 0..infinity)	(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]\[Nu]) BesselY[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == Divide[1,Pi]Integrate[Exp[\[Zeta]Cos[\[Theta]]]Sin[zSin[\[Theta]]- \[Nu]\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[1,Pi]Integrate[(Exp[\[Nu]t + \[Zeta]Cosh[t]]+ Exp[- \[Nu]t - \[Zeta]Cosh[t]]Cos[\[Nu]Pi])* Exp[- z*Sinh[t]], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E15	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}{H^{(1)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi i% }e^{-\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t+i\zeta\sinh t-\nu t% }\mathrm{d}t}}$ `\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\HankelH{1}{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = \frac{1}{\pi i}e^{-\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{iz\cosh@@{t}+i\zeta\sinh@@{t}-\nu t}\diff{t}`		((z + zeta)/(z - zeta))^((1)/(2)nu) HankelH1(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = (1)/(PiI)exp(-(1)/(2)nuPiI)int(exp(Izcosh(t)+ Izetasinh(t)- nu*t), t = - infinity..infinity)	(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]\[Nu]) HankelH1[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == Divide[1,PiI]Exp[-Divide[1,2]\[Nu]PiI]Integrate[Exp[IzCosh[t]+ I\[Zeta]Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E16	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}{H^{(2)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i% }e^{\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t% }\mathrm{d}t}}$ `\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\HankelH{2}{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = -\frac{1}{\pi i}e^{\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh@@{t}-i\zeta\sinh@@{t}-\nu t}\diff{t}`	${\displaystyle{\displaystyle\Im\left(z+\zeta\right)<0,\Im\left(z-\zeta\right)<% 0}}$	((z + zeta)/(z - zeta))^((1)/(2)nu) HankelH2(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = -(1)/(PiI)exp((1)/(2)nuPiI)int(exp(- Izcosh(t)- Izetasinh(t)- nu*t), t = - infinity..infinity)	(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]\[Nu]) HankelH2[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == -Divide[1,PiI]Exp[Divide[1,2]\[Nu]PiI]Integrate[Exp[- IzCosh[t]- I\[Zeta]Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E17	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{% \infty-\pi i}^{\infty+\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}$ `\BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, z) = (1)/(2PiI)int(exp(zsinh(t)- nut), t = infinity - PiI..infinity + Pi*I)	BesselJ[\[Nu], z] == Divide[1,2PiI]Integrate[Exp[zSinh[t]- \[Nu]t], {t, Infinity - PiI, Infinity + Pi*I}, GenerateConditions->None]	Error	Failure	-	Failed [70 / 70] Result: Complex[0.4358908643715884, -0.07192294931339177] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.0679098760861825, 0.09257666026367889] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.9#Ex7	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{1}{\pi i}\int_% {-\infty}^{\infty+\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}$ `\HankelH{1}{\nu}@{z} = \frac{1}{\pi i}\int_{-\infty}^{\infty+\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}`		HankelH1(nu, z) = (1)/(PiI)int(exp(zsinh(t)- nut), t = - infinity..infinity + Pi*I)	HankelH1[\[Nu], z] == Divide[1,PiI]Integrate[Exp[zSinh[t]- \[Nu]t], {t, - Infinity, Infinity + Pi*I}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9#Ex8	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{1}{\pi i}\int% _{-\infty}^{\infty-\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}$ `\HankelH{2}{\nu}@{z} = -\frac{1}{\pi i}\int_{-\infty}^{\infty-\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}`		HankelH2(nu, z) = -(1)/(PiI)int(exp(zsinh(t)- nut), t = - infinity..infinity - Pi*I)	HankelH2[\[Nu], z] == -Divide[1,PiI]Integrate[Exp[zSinh[t]- \[Nu]t], {t, - Infinity, Infinity - Pi*I}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.9.E19	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{2\pi i}\int_{-\infty}^{(0+)}\exp\left(t-\frac{z^{2}}{4t}\right)\frac{\mathrm% {d}t}{t^{\nu+1}}}}$ `\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{2\pi i}\int_{-\infty}^{(0+)}\exp@{t-\frac{z^{2}}{4t}}\frac{\diff{t}}{t^{\nu+1}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, z) = (((1)/(2)z)^(nu))/(2PiI)int(exp(t -((z)^(2))/(4t))(1)/((t)^(nu + 1)), t = - infinity..(0 +))	BesselJ[\[Nu], z] == Divide[(Divide[1,2]z)^\[Nu],2PiI]Integrate[Exp[t -Divide[(z)^(2),4t]]Divide[1,(t)^(\[Nu]+ 1)], {t, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
10.9.E20	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{% 2}-\nu\right)(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left% (zt\right)(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}$ `\BesselJ{\nu}@{z} = \frac{\EulerGamma@{\frac{1}{2}-\nu}(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos@{zt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}`	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\frac{1}{2}% -\nu)>0}}$	BesselJ(nu, z) = (GAMMA((1)/(2)- nu)((1)/(2)z)^(nu))/((Pi)^((3)/(2))* I)int(cos(zt)*((t)^(2)- 1)^(nu -(1)/(2)), t = 0..(1 +))	BesselJ[\[Nu], z] == Divide[Gamma[Divide[1,2]- \[Nu]](Divide[1,2]z)^\[Nu],(Pi)^(Divide[3,2])* I]Integrate[Cos[zt]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 0, (1 +)}, GenerateConditions->None]	Error	Failure	-	Error
10.9#Ex9	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{\Gamma\left(% \tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1+i% \infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}$ `\HankelH{1}{\nu}@{z} = \frac{\EulerGamma@{\tfrac{1}{2}-\nu}(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1+i\infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}`	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2}<\tfrac{1}{2}\pi,% \ldots<\tfrac{1}{2}\pi,\|\operatorname{ph}z\|<\tfrac{1}{2}\pi,\Re(\tfrac{1}{2}-% \nu)>0}}$	HankelH1(nu, z) = (GAMMA((1)/(2)- nu)((1)/(2)z)^(nu))/((Pi)^((3)/(2))* I)int(exp(Izt)((t)^(2)- 1)^(nu -(1)/(2)), t = 1 + I*infinity..(1 +))	HankelH1[\[Nu], z] == Divide[Gamma[Divide[1,2]- \[Nu]](Divide[1,2]z)^\[Nu],(Pi)^(Divide[3,2])* I]Integrate[Exp[Izt]((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1 + I*Infinity, (1 +)}, GenerateConditions->None]	Error	Failure	-	Error
10.9#Ex10	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=\frac{\Gamma\left(% \tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1-i% \infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}$ `\HankelH{2}{\nu}@{z} = \frac{\EulerGamma@{\tfrac{1}{2}-\nu}(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1-i\infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}`	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2}<\tfrac{1}{2}\pi,% \ldots<\tfrac{1}{2}\pi,\|\operatorname{ph}z\|<\tfrac{1}{2}\pi,\Re(\tfrac{1}{2}-% \nu)>0}}$	HankelH2(nu, z) = (GAMMA((1)/(2)- nu)((1)/(2)z)^(nu))/((Pi)^((3)/(2))* I)int(exp(- Izt)((t)^(2)- 1)^(nu -(1)/(2)), t = 1 - I*infinity..(1 +))	HankelH2[\[Nu], z] == Divide[Gamma[Divide[1,2]- \[Nu]](Divide[1,2]z)^\[Nu],(Pi)^(Divide[3,2])* I]Integrate[Exp[- Izt]((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1 - I*Infinity, (1 +)}, GenerateConditions->None]	Error	Failure	-	Error
10.9.E22	${\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i% \infty}^{i\infty}\frac{\Gamma\left(-t\right)(\tfrac{1}{2}x)^{\nu+2t}}{\Gamma% \left(\nu+t+1\right)}\mathrm{d}t}}$ `\BesselJ{\nu}@{x} = \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\EulerGamma@{-t}(\tfrac{1}{2}x)^{\nu+2t}}{\EulerGamma@{\nu+t+1}}\diff{t}`	${\displaystyle{\displaystyle\Re\nu>0,x>0,\Re(\nu+k+1)>0,\Re(-t)>0,\Re(\nu+t+1)% >0}}$	BesselJ(nu, x) = (1)/(2PiI)int((GAMMA(- t)((1)/(2)x)^(nu + 2t))/(GAMMA(nu + t + 1)), t = - Iinfinity..Iinfinity)	BesselJ[\[Nu], x] == Divide[1,2PiI]Integrate[Divide[Gamma[- t](Divide[1,2]x)^(\[Nu]+ 2t),Gamma[\[Nu]+ t + 1]], {t, - IInfinity, IInfinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E23	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{-% \infty-ic}^{-\infty+ic}\frac{\Gamma\left(t\right)}{\Gamma\left(\nu-t+1\right)}% (\tfrac{1}{2}z)^{\nu-2t}\mathrm{d}t}}$ `\BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{-\infty-ic}^{-\infty+ic}\frac{\EulerGamma@{t}}{\EulerGamma@{\nu-t+1}}(\tfrac{1}{2}z)^{\nu-2t}\diff{t}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re t>0,\Re(\nu-t+1)>0}}$	BesselJ(nu, z) = (1)/(2PiI)int((GAMMA(t))/(GAMMA(nu - t + 1))((1)/(2)z)^(nu - 2t), t = - infinity - Ic..- infinity + Ic)	BesselJ[\[Nu], z] == Divide[1,2PiI]Integrate[Divide[Gamma[t],Gamma[\[Nu]- t + 1]](Divide[1,2]z)^(\[Nu]- 2t), {t, - Infinity - Ic, - Infinity + Ic}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [300 / 300] Result: Complex[0.4358908643715884, -0.07192294931339177] Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.0679098760861825, 0.09257666026367889] Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.9.E24	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=-\frac{e^{-\frac{1}{% 2}\nu\pi i}}{2\pi^{2}}\\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma% \left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t}}$ `\HankelH{1}{\nu}@{z} = -\frac{e^{-\frac{1}{2}\nu\pi i}}{2\pi^{2}}\\int_{c-i\infty}^{c+i\infty}\EulerGamma@{t}\EulerGamma@{t-\nu}(-\tfrac{1}{2}iz)^{\nu-2t}\diff{t}`	${\displaystyle{\displaystyle 0<\operatorname{ph}z,\operatorname{ph}z<\pi,\Re t% >0,\Re(t-\nu)>0}}$	HankelH1(nu, z) = -(exp(-(1)/(2)nuPiI))/(2(Pi)^(2))* int(GAMMA(t)GAMMA(t - nu)(-(1)/(2)Iz)^(nu - 2t), t = c - Iinfinity..c + I*infinity)	HankelH1[\[Nu], z] == -Divide[Exp[-Divide[1,2]\[Nu]PiI],2(Pi)^(2)]* Integrate[Gamma[t]Gamma[t - \[Nu]](-Divide[1,2]Iz)^(\[Nu]- 2t), {t, c - IInfinity, c + I*Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [120 / 120] Result: .2971181619-.8401954886I Test Values: {c = -3/2, nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -.8661908042+.2691615148I Test Values: {c = -3/2, nu = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
10.9.E25	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=\frac{e^{\frac{1}{2}% \nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma% \left(t-\nu\right)(\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t}}$ `\HankelH{2}{\nu}@{z} = \frac{e^{\frac{1}{2}\nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{t}\EulerGamma@{t-\nu}(\tfrac{1}{2}iz)^{\nu-2t}\diff{t}`	${\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<0,\Re t% >0,\Re(t-\nu)>0}}$	HankelH2(nu, z) = (exp((1)/(2)nuPiI))/(2(Pi)^(2))int(GAMMA(t)GAMMA(t - nu)((1)/(2)Iz)^(nu - 2t), t = c - Iinfinity..c + Iinfinity)	HankelH2[\[Nu], z] == Divide[Exp[Divide[1,2]\[Nu]PiI],2(Pi)^(2)]Integrate[Gamma[t]Gamma[t - \[Nu]](Divide[1,2]Iz)^(\[Nu]- 2t), {t, c - IInfinity, c + IInfinity}, GenerateConditions->None]	Failure	Aborted	Failed [120 / 120] Result: -.1414870617+.1246394392I Test Values: {c = -3/2, nu = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: -.1498748781e-1-.1846515642I Test Values: {c = -3/2, nu = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Skipped - Because timed out
10.9.E26	${\displaystyle{\displaystyle J_{\mu}\left(z\right)J_{\nu}\left(z\right)=\frac{% 2}{\pi}\int_{0}^{\pi/2}J_{\mu+\nu}\left(2z\cos\theta\right)\cos\left((\mu-\nu)% \theta\right)\mathrm{d}\theta}}$ `\BesselJ{\mu}@{z}\BesselJ{\nu}@{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\BesselJ{\mu+\nu}@{2z\cos@@{\theta}}\cos@{(\mu-\nu)\theta}\diff{\theta}`	${\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>-1,\Re((\mu)+k+1)>0,\Re(% \nu+k+1)>0,\Re((\mu+\nu)+k+1)>0}}$	BesselJ(mu, z)BesselJ(nu, z) = (2)/(Pi)int(BesselJ(mu + nu, 2zcos(theta))cos((mu - nu)theta), theta = 0..Pi/2)	BesselJ[\[Mu], z]BesselJ[\[Nu], z] == Divide[2,Pi]Integrate[BesselJ[\[Mu]+ \[Nu], 2zCos[\[Theta]]]Cos[(\[Mu]- \[Nu])\[Theta]], {\[Theta], 0, Pi/2}, GenerateConditions->None]	Failure	Aborted	Manual Skip!	Skipped - Because timed out
10.9.E27	${\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=% \frac{2}{\pi}\int_{0}^{\pi/2}J_{2\nu}\left(2(z\zeta)^{\frac{1}{2}}\sin\theta% \right)\cos\left((z-\zeta)\cos\theta\right)\mathrm{d}\theta}}$ `\BesselJ{\nu}@{z}\BesselJ{\nu}@{\zeta} = \frac{2}{\pi}\int_{0}^{\pi/2}\BesselJ{2\nu}@{2(z\zeta)^{\frac{1}{2}}\sin@@{\theta}}\cos@{(z-\zeta)\cos@@{\theta}}\diff{\theta}`	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((2\nu)+k+1% )>0}}$	BesselJ(nu, z)BesselJ(nu, zeta) = (2)/(Pi)int(BesselJ(2nu, 2(zzeta)^((1)/(2)) sin(theta))cos((z - zeta)cos(theta)), theta = 0..Pi/2)	BesselJ[\[Nu], z]BesselJ[\[Nu], \[Zeta]] == Divide[2,Pi]Integrate[BesselJ[2\[Nu], 2(z\[Zeta])^(Divide[1,2]) Sin[\[Theta]]]Cos[(z - \[Zeta])Cos[\[Theta]]], {\[Theta], 0, Pi/2}, GenerateConditions->None]	Failure	Aborted	Manual Skip!	Skipped - Because timed out
10.9.E28	${\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=% \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\\exp\left(\frac{1}{2}t-\frac{z^{% 2}+\zeta^{2}}{2t}\right)I_{\nu}\left(\frac{z\zeta}{t}\right)\frac{\mathrm{d}t}% {t}}}$ `\BesselJ{\nu}@{z}\BesselJ{\nu}@{\zeta} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\\exp@{\frac{1}{2}t-\frac{z^{2}+\zeta^{2}}{2t}}\modBesselI{\nu}@{\frac{z\zeta}{t}}\frac{\diff{t}}{t}`	${\displaystyle{\displaystyle\Re\nu>-1,\Re(\nu+k+1)>0}}$	BesselJ(nu, z)BesselJ(nu, zeta) = (1)/(2PiI)int(* exp((1)/(2)t -((z)^(2)+ (zeta)^(2))/(2t))BesselI(nu, (zzeta)/(t))(1)/(t), t = c - Iinfinity..c + I*infinity)	BesselJ[\[Nu], z]BesselJ[\[Nu], \[Zeta]] == Divide[1,2PiI]Integrate[* Exp[Divide[1,2]t -Divide[(z)^(2)+ \[Zeta]^(2),2t]]BesselI[\[Nu], Divide[z\[Zeta],t]]Divide[1,t], {t, c - IInfinity, c + I*Infinity}, GenerateConditions->None]	Error	Failure	-	Error
10.9.E29	${\displaystyle{\displaystyle J_{\mu}\left(x\right)J_{\nu}\left(x\right)=\frac{% 1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+% \mu+\nu+1\right)(\tfrac{1}{2}x)^{\mu+\nu+2t}}{\Gamma\left(t+\mu+1\right)\Gamma% \left(t+\nu+1\right)\Gamma\left(t+\mu+\nu+1\right)}\mathrm{d}t}}$ `\BesselJ{\mu}@{x}\BesselJ{\nu}@{x} = \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\EulerGamma@{-t}\EulerGamma@{2t+\mu+\nu+1}(\tfrac{1}{2}x)^{\mu+\nu+2t}}{\EulerGamma@{t+\mu+1}\EulerGamma@{t+\nu+1}\EulerGamma@{t+\mu+\nu+1}}\diff{t}`	${\displaystyle{\displaystyle x>0,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,\Re(-t)>0,\Re% (2t+\mu+\nu+1)>0,\Re(t+\mu+1)>0,\Re(t+\nu+1)>0,\Re(t+\mu+\nu+1)>0}}$	BesselJ(mu, x)BesselJ(nu, x) = (1)/(2PiI)int((GAMMA(- t)GAMMA(2t + mu + nu + 1)((1)/(2)x)^(mu + nu + 2t))/(GAMMA(t + mu + 1)GAMMA(t + nu + 1)GAMMA(t + mu + nu + 1)), t = - Iinfinity..I*infinity)	BesselJ[\[Mu], x]BesselJ[\[Nu], x] == Divide[1,2PiI]Integrate[Divide[Gamma[- t]Gamma[2t + \[Mu]+ \[Nu]+ 1](Divide[1,2]x)^(\[Mu]+ \[Nu]+ 2t),Gamma[t + \[Mu]+ 1]Gamma[t + \[Nu]+ 1]Gamma[t + \[Mu]+ \[Nu]+ 1]], {t, - IInfinity, I*Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.9.E30	${\displaystyle{\displaystyle{J_{\nu}^{2}}\left(z\right)+{Y_{\nu}^{2}}\left(z% \right)=\frac{8}{\pi^{2}}\int_{0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2% z\sinh t\right)\mathrm{d}t}}$ `\BesselJ{\nu}^{2}@{z}+\BesselY{\nu}^{2}@{z} = \frac{8}{\pi^{2}}\int_{0}^{\infty}\cosh@{2\nu t}\modBesselK{0}@{2z\sinh@@{t}}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0,\Re((-\nu)+k+1)>0}}$	(BesselJ(nu, z))^(2)+ (BesselY(nu, z))^(2) = (8)/((Pi)^(2))int(cosh(2nut)BesselK(0, 2zsinh(t)), t = 0..infinity)	(BesselJ[\[Nu], z])^(2)+ (BesselY[\[Nu], z])^(2) == Divide[8,(Pi)^(2)]Integrate[Cosh[2\[Nu]t]BesselK[0, 2zSinh[t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.11.E1	${\displaystyle{\displaystyle J_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}J_{% \nu}\left(z\right)}}$ `\BesselJ{\nu}@{ze^{m\pi i}} = e^{m\nu\pi i}\BesselJ{\nu}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, zexp(mPiI)) = exp(mnuPiI)*BesselJ(nu, z)	BesselJ[\[Nu], zExp[mPiI]] == Exp[m\[Nu]PiI]*BesselJ[\[Nu], z]	Failure	Failure	Failed [132 / 210] Result: -1.978604450-.5916012221I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 1} Result: .4256613630-.5580360922e-1I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 2} ... skip entries to safe data	Failed [120 / 210] Result: Complex[-1.9786044502778974, -0.5916012230349773] Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.42566136315461117, -0.05580360945599949] Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11.E2	${\displaystyle{\displaystyle Y_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}Y_{% \nu}\left(z\right)+2i\sin\left(m\nu\pi\right)\cot\left(\nu\pi\right)J_{\nu}% \left(z\right)}}$ `\BesselY{\nu}@{ze^{m\pi i}} = e^{-m\nu\pi i}\BesselY{\nu}@{z}+2i\sin@{m\nu\pi}\cot@{\nu\pi}\BesselJ{\nu}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, zexp(mPiI)) = exp(- mnuPiI)BesselY(nu, z)+ 2Isin(mnuPi)cot(nuPi)BesselJ(nu, z)	BesselY[\[Nu], zExp[mPiI]] == Exp[- m\[Nu]PiI]BesselY[\[Nu], z]+ 2ISin[m\[Nu]Pi]Cot[\[Nu]Pi]BesselJ[\[Nu], z]	Failure	Failure	Failed [170 / 210] Result: -4.492502702+3.271310776I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 1} Result: 19.72399963+2.416868418I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 2} ... skip entries to safe data	Failed [162 / 210] Result: Complex[-4.49250270148862, 3.2713107749000305] Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[19.723999620348792, 2.416868461226219] Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11.E3	${\displaystyle{\displaystyle\sin\left(\nu\pi\right){H^{(1)}_{\nu}}\left(ze^{m% \pi i}\right)=-\sin\left((m-1)\nu\pi\right){H^{(1)}_{\nu}}\left(z\right)-e^{-% \nu\pi i}\sin\left(m\nu\pi\right){H^{(2)}_{\nu}}\left(z\right)}}$ `\sin@{\nu\pi}\HankelH{1}{\nu}@{ze^{m\pi i}} = -\sin@{(m-1)\nu\pi}\HankelH{1}{\nu}@{z}-e^{-\nu\pi i}\sin@{m\nu\pi}\HankelH{2}{\nu}@{z}`		sin(nuPi)HankelH1(nu, zexp(mPiI)) = - sin((m - 1)nuPi)HankelH1(nu, z)- exp(- nuPiI)sin(mnuPi)HankelH2(nu, z)	Sin[\[Nu]Pi]HankelH1[\[Nu], zExp[mPiI]] == - Sin[(m - 1)\[Nu]Pi]HankelH1[\[Nu], z]- Exp[- \[Nu]PiI]Sin[m\[Nu]Pi]HankelH2[\[Nu], z]	Failure	Failure	Failed [132 / 210] Result: -16.06107638+5.815014709I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 1} Result: 39.27071892+24.34608468I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 2} ... skip entries to safe data	Failed [120 / 210] Result: Complex[-16.061076381218605, 5.815014694873561] Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[39.27071883811536, 24.346084784539414] Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11.E4	${\displaystyle{\displaystyle\sin\left(\nu\pi\right){H^{(2)}_{\nu}}\left(ze^{m% \pi i}\right)=e^{\nu\pi i}\sin\left(m\nu\pi\right){H^{(1)}_{\nu}}\left(z\right% )+\sin\left((m+1)\nu\pi\right){H^{(2)}_{\nu}}\left(z\right)}}$ `\sin@{\nu\pi}\HankelH{2}{\nu}@{ze^{m\pi i}} = e^{\nu\pi i}\sin@{m\nu\pi}\HankelH{1}{\nu}@{z}+\sin@{(m+1)\nu\pi}\HankelH{2}{\nu}@{z}`		sin(nuPi)HankelH2(nu, zexp(mPiI)) = exp(nuPiI)sin(mnuPi)HankelH1(nu, z)+ sin((m + 1)nuPi)HankelH2(nu, z)	Sin[\[Nu]Pi]HankelH2[\[Nu], zExp[mPiI]] == Exp[\[Nu]PiI]Sin[m\[Nu]Pi]HankelH1[\[Nu], z]+ Sin[(m + 1)\[Nu]Pi]HankelH2[\[Nu], z]	Failure	Failure	Failed [132 / 210] Result: 9.518923666+1.283901315I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 1} Result: -38.63237633-26.24866521I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, m = 2} ... skip entries to safe data	Failed [120 / 210] Result: Complex[9.518923662743454, 1.2839013369012835] Test Values: {Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-38.63237622058036, -26.24866530437453] Test Values: {Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11#Ex1	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(ze^{\pi i}\right)=-e^{-\nu\pi i% }{H^{(2)}_{\nu}}\left(z\right)}}$ `\HankelH{1}{\nu}@{ze^{\pi i}} = -e^{-\nu\pi i}\HankelH{2}{\nu}@{z}`		HankelH1(nu, zexp(PiI)) = - exp(- nuPiI)*HankelH2(nu, z)	HankelH1[\[Nu], zExp[PiI]] == - Exp[- \[Nu]PiI]*HankelH2[\[Nu], z]	Failure	Failure	Failed [20 / 70] Result: -5.249915228-5.084103922I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I} Result: -3.129030441-5.176244122I Test Values: {nu = 1/23^(1/2)+1/2I, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [20 / 70] Result: Complex[-5.2499152251779275, -5.084103924523598] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.4609763579335797, 35.01102127779514] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.11#Ex2	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(ze^{-\pi i}\right)=-e^{\nu\pi i% }{H^{(1)}_{\nu}}\left(z\right)}}$ `\HankelH{2}{\nu}@{ze^{-\pi i}} = -e^{\nu\pi i}\HankelH{1}{\nu}@{z}`		HankelH2(nu, zexp(- PiI)) = - exp(nuPiI)*HankelH1(nu, z)	HankelH2[\[Nu], zExp[- PiI]] == - Exp[\[Nu]PiI]*HankelH1[\[Nu], z]	Failure	Failure	Failed [50 / 70] Result: 1.033334476+.7163604616I Test Values: {nu = 1/23^(1/2)+1/2I, z = 1/2-1/2I3^(1/2)} Result: 1.427918302+.5187414665I Test Values: {nu = 1/23^(1/2)+1/2I, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Failed [50 / 70] Result: Complex[1.0333344760783634, 0.7163604618419928] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.538721989873022, -0.29666827540401164] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.11.E6	${\displaystyle{\displaystyle Y_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}(Y_{n}% \left(z\right)+2imJ_{n}\left(z\right))}}$ `\BesselY{n}@{ze^{m\pi i}} = (-1)^{mn}(\BesselY{n}@{z}+2im\BesselJ{n}@{z})`	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re((-n)+k+1)>0}}$	BesselY(n, zexp(mPiI)) = (- 1)^(mn)(BesselY(n, z)+ 2ImBesselJ(n, z))	BesselY[n, zExp[mPiI]] == (- 1)^(mn)(BesselY[n, z]+ 2ImBesselJ[n, z])	Failure	Failure	Failed [57 / 63] Result: -.7553141392+1.723217630I Test Values: {z = 1/23^(1/2)+1/2I, m = 1, n = 1} Result: .3969469092-.2695422112I Test Values: {z = 1/23^(1/2)+1/2I, m = 1, n = 2} ... skip entries to safe data	Failed [48 / 63] Result: Complex[-0.7553141389736522, 1.7232176296930342] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.39694690825884216, -0.26954221211204654] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11.E7	${\displaystyle{\displaystyle{H^{(1)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn-1}(% (m-1){H^{(1)}_{n}}\left(z\right)+m{H^{(2)}_{n}}\left(z\right))}}$ `\HankelH{1}{n}@{ze^{m\pi i}} = (-1)^{mn-1}((m-1)\HankelH{1}{n}@{z}+m\HankelH{2}{n}@{z})`		HankelH1(n, zexp(mPiI)) = (- 1)^(mn - 1)((m - 1)HankelH1(n, z)+ m*HankelH2(n, z))	HankelH1[n, zExp[mPiI]] == (- 1)^(mn - 1)((m - 1)HankelH1[n, z]+ m*HankelH2[n, z])	Failure	Failure	Failed [57 / 63] Result: -1.723217630-.7553141394I Test Values: {z = 1/23^(1/2)+1/2I, m = 1, n = 1} Result: .2695422111+.3969469092I Test Values: {z = 1/23^(1/2)+1/2I, m = 1, n = 2} ... skip entries to safe data	Failed [48 / 63] Result: Complex[-1.7232176296930342, -0.7553141389736522] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.26954221211204654, 0.39694690825884216] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11.E8	${\displaystyle{\displaystyle{H^{(2)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn}(m{% H^{(1)}_{n}}\left(z\right)+(m+1){H^{(2)}_{n}}\left(z\right))}}$ `\HankelH{2}{n}@{ze^{m\pi i}} = (-1)^{mn}(m\HankelH{1}{n}@{z}+(m+1)\HankelH{2}{n}@{z})`		HankelH2(n, zexp(mPiI)) = (- 1)^(mn)(mHankelH1(n, z)+(m + 1)*HankelH2(n, z))	HankelH2[n, zExp[mPiI]] == (- 1)^(mn)(mHankelH1[n, z]+(m + 1)*HankelH2[n, z])	Failure	Failure	Failed [57 / 63] Result: 1.723217630+.755314139I Test Values: {z = 1/23^(1/2)+1/2I, m = 1, n = 1} Result: -.269542211-.396946909I Test Values: {z = 1/23^(1/2)+1/2I, m = 1, n = 2} ... skip entries to safe data	Failed [48 / 63] Result: Complex[1.7232176296930342, 0.7553141389736524] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.26954221211204654, -0.39694690825884216] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.11#E9X	${\displaystyle{\displaystyle\displaystyle J_{\nu}\left(\overline{z}\right)=% \overline{J_{\nu}\left(z\right)}}}$ `\displaystyle\BesselJ{\nu}@{\conj{z}} = \conj{\BesselJ{\nu}@{z}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, conjugate(z)) = conjugate(BesselJ(nu, z))	BesselJ[\[Nu], Conjugate[z]] == Conjugate[BesselJ[\[Nu], z]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.11#E9X	${\displaystyle{\displaystyle\displaystyle Y_{\nu}\left(\overline{z}\right)=% \overline{Y_{\nu}\left(z\right)}}}$ `\displaystyle\BesselY{\nu}@{\conj{z}} = \conj{\BesselY{\nu}@{z}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, conjugate(z)) = conjugate(BesselY(nu, z))	BesselY[\[Nu], Conjugate[z]] == Conjugate[BesselY[\[Nu], z]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.11#E9Xa	${\displaystyle{\displaystyle\displaystyle{H^{(1)}_{\nu}}\left(\overline{z}% \right)=\overline{{H^{(2)}_{\nu}}\left(z\right)}}}$ `\displaystyle\HankelH{1}{\nu}@{\conj{z}} = \conj{\HankelH{2}{\nu}@{z}}`		HankelH1(nu, conjugate(z)) = conjugate(HankelH2(nu, z))	HankelH1[\[Nu], Conjugate[z]] == Conjugate[HankelH2[\[Nu], z]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.11#E9Xa	${\displaystyle{\displaystyle\displaystyle{H^{(2)}_{\nu}}\left(\overline{z}% \right)=\overline{{H^{(1)}_{\nu}}\left(z\right)}}}$ `\displaystyle\HankelH{2}{\nu}@{\conj{z}} = \conj{\HankelH{1}{\nu}@{z}}`		HankelH2(nu, conjugate(z)) = conjugate(HankelH1(nu, z))	HankelH2[\[Nu], Conjugate[z]] == Conjugate[HankelH1[\[Nu], z]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.12.E1	${\displaystyle{\displaystyle e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{% \infty}t^{m}J_{m}\left(z\right)}}$ `e^{\frac{1}{2}z(t-t^{-1})} = \sum_{m=-\infty}^{\infty}t^{m}\BesselJ{m}@{z}`	${\displaystyle{\displaystyle\Re(m+k+1)>0}}$	exp((1)/(2)z(t - (t)^(- 1))) = sum((t)^(m)* BesselJ(m, z), m = - infinity..infinity)	Exp[Divide[1,2]z(t - (t)^(- 1))] == Sum[(t)^(m)* BesselJ[m, z], {m, - Infinity, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 42]	Successful [Tested: 42]
10.12#Ex1	${\displaystyle{\displaystyle\cos\left(z\sin\theta\right)=J_{0}\left(z\right)+2% \sum_{k=1}^{\infty}J_{2k}\left(z\right)\cos\left(2k\theta\right)}}$ `\cos@{z\sin@@{\theta}} = \BesselJ{0}@{z}+2\sum_{k=1}^{\infty}\BesselJ{2k}@{z}\cos@{2k\theta}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0}}$	cos(zsin(theta)) = BesselJ(0, z)+ 2sum(BesselJ(2k, z)cos(2ktheta), k = 1..infinity)	Cos[zSin[\[Theta]]] == BesselJ[0, z]+ 2Sum[BesselJ[2k, z]Cos[2k\[Theta]], {k, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Skipped - Because timed out	Successful [Tested: 70]
10.12#Ex2	${\displaystyle{\displaystyle\sin\left(z\sin\theta\right)=2\sum_{k=0}^{\infty}J% _{2k+1}\left(z\right)\sin\left((2k+1)\theta\right)}}$ `\sin@{z\sin@@{\theta}} = 2\sum_{k=0}^{\infty}\BesselJ{2k+1}@{z}\sin@{(2k+1)\theta}`	${\displaystyle{\displaystyle\Re((2k+1)+k+1)>0}}$	sin(zsin(theta)) = 2sum(BesselJ(2k + 1, z)sin((2k + 1)theta), k = 0..infinity)	Sin[zSin[\[Theta]]] == 2Sum[BesselJ[2k + 1, z]Sin[(2k + 1)\[Theta]], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Successful	Skipped - Because timed out	Successful [Tested: 70]
10.12#Ex3	${\displaystyle{\displaystyle\cos\left(z\cos\theta\right)=J_{0}\left(z\right)+2% \sum_{k=1}^{\infty}(-1)^{k}J_{2k}\left(z\right)\cos\left(2k\theta\right)}}$ `\cos@{z\cos@@{\theta}} = \BesselJ{0}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{2k}@{z}\cos@{2k\theta}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0}}$	cos(zcos(theta)) = BesselJ(0, z)+ 2sum((- 1)^(k)* BesselJ(2k, z)cos(2ktheta), k = 1..infinity)	Cos[zCos[\[Theta]]] == BesselJ[0, z]+ 2Sum[(- 1)^(k)* BesselJ[2k, z]Cos[2k\[Theta]], {k, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Skipped - Because timed out	Successful [Tested: 70]
10.12#Ex4	${\displaystyle{\displaystyle\sin\left(z\cos\theta\right)=2\sum_{k=0}^{\infty}(% -1)^{k}J_{2k+1}\left(z\right)\cos\left((2k+1)\theta\right)}}$ `\sin@{z\cos@@{\theta}} = 2\sum_{k=0}^{\infty}(-1)^{k}\BesselJ{2k+1}@{z}\cos@{(2k+1)\theta}`	${\displaystyle{\displaystyle\Re((2k+1)+k+1)>0}}$	sin(zcos(theta)) = 2sum((- 1)^(k)* BesselJ(2k + 1, z)cos((2k + 1)theta), k = 0..infinity)	Sin[zCos[\[Theta]]] == 2Sum[(- 1)^(k)* BesselJ[2k + 1, z]Cos[(2k + 1)\[Theta]], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Successful	Skipped - Because timed out	Successful [Tested: 70]
10.12.E4	${\displaystyle{\displaystyle 1=J_{0}\left(z\right)+2J_{2}\left(z\right)+2J_{4}% \left(z\right)+2J_{6}\left(z\right)+\cdots}}$ `1 = \BesselJ{0}@{z}+2\BesselJ{2}@{z}+2\BesselJ{4}@{z}+2\BesselJ{6}@{z}+\dotsb`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(2+k+1)>0,\Re(4+k+1)>0,\Re(6+k+1)>% 0}}$	1 = BesselJ(0, z)+ 2BesselJ(2, z)+ 2BesselJ(4, z)+ 2*BesselJ(6, z)+ ..	1 == BesselJ[0, z]+ 2BesselJ[2, z]+ 2BesselJ[4, z]+ 2*BesselJ[6, z]+ \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-9.924736618779559^-8, -1.6360842739013975^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-9.440290587615918^-8, -1.7199789187696823^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.12#Ex5	${\displaystyle{\displaystyle\cos z=J_{0}\left(z\right)-2J_{2}\left(z\right)+2J% _{4}\left(z\right)-2J_{6}\left(z\right)+\cdots}}$ `\cos@@{z} = \BesselJ{0}@{z}-2\BesselJ{2}@{z}+2\BesselJ{4}@{z}-2\BesselJ{6}@{z}+\dotsb`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(2+k+1)>0,\Re(4+k+1)>0,\Re(6+k+1)>% 0}}$	cos(z) = BesselJ(0, z)- 2BesselJ(2, z)+ 2BesselJ(4, z)- 2*BesselJ(6, z)+ ..	Cos[z] == BesselJ[0, z]- 2BesselJ[2, z]+ 2BesselJ[4, z]- 2*BesselJ[6, z]+ \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-9.976125969757277^-8, -1.6267640928768756^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-9.384008414770051^-8, -1.7292990711625933^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.12#Ex6	${\displaystyle{\displaystyle\sin z=2J_{1}\left(z\right)-2J_{3}\left(z\right)+2% J_{5}\left(z\right)-\cdots}}$ `\sin@@{z} = 2\BesselJ{1}@{z}-2\BesselJ{3}@{z}+2\BesselJ{5}@{z}-\dotsb`	${\displaystyle{\displaystyle\Re(1+k+1)>0,\Re(3+k+1)>0,\Re(5+k+1)>0}}$	sin(z) = 2BesselJ(1, z)- 2BesselJ(3, z)+ 2*BesselJ(5, z)- ..	Sin[z] == 2BesselJ[1, z]- 2BesselJ[3, z]+ 2*BesselJ[5, z]- \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[2.683443869444524^-6, 1.443280323643048^-6], …] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[1.6585570595806232^-6, -2.68341820086615^-6], …] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.12#Ex7	${\displaystyle{\displaystyle\tfrac{1}{2}z\cos z=J_{1}\left(z\right)-9J_{3}% \left(z\right)+25J_{5}\left(z\right)-49J_{7}\left(z\right)+\cdots}}$ `\tfrac{1}{2}z\cos@@{z} = \BesselJ{1}@{z}-9\BesselJ{3}@{z}+25\BesselJ{5}@{z}-49\BesselJ{7}@{z}+\dotsb`	${\displaystyle{\displaystyle\Re(1+k+1)>0,\Re(3+k+1)>0,\Re(5+k+1)>0,\Re(7+k+1)>% 0}}$	(1)/(2)zcos(z) = BesselJ(1, z)- 9BesselJ(3, z)+ 25BesselJ(5, z)- 49*BesselJ(7, z)+ ..	Divide[1,2]zCos[z] == BesselJ[1, z]- 9BesselJ[3, z]+ 25BesselJ[5, z]- 49*BesselJ[7, z]+ \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-1.0583928733431947^-8, -4.2969798588234076^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[4.4207480831559565^-7, 1.0857586385526474^-8], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.12#Ex8	${\displaystyle{\displaystyle\tfrac{1}{2}z\sin z=4J_{2}\left(z\right)-16J_{4}% \left(z\right)+36J_{6}\left(z\right)-\cdots}}$ `\tfrac{1}{2}z\sin@@{z} = 4\BesselJ{2}@{z}-16\BesselJ{4}@{z}+36\BesselJ{6}@{z}-\dotsi`	${\displaystyle{\displaystyle\Re(2+k+1)>0,\Re(4+k+1)>0,\Re(6+k+1)>0}}$	(1)/(2)zsin(z) = 4BesselJ(2, z)- 16BesselJ(4, z)+ 36*BesselJ(6, z)- ..	Divide[1,2]zSin[z] == 4BesselJ[2, z]- 16BesselJ[4, z]+ 36*BesselJ[6, z]- \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[3.196945008165919^-6, 5.1972576656234^-6], …] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[2.997776089863624^-6, 5.542144419168338^-6], …] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.13.E8	${\displaystyle{\displaystyle w^{(2n)}=(-1)^{n}\lambda^{2n}z^{-n}w}}$ `w^{(2n)} = (-1)^{n}\lambda^{2n}z^{-n}w`	${\displaystyle{\displaystyle w=z^{\frac{1}{2}n}\mathscr{C}_{n}\left(2\lambda e% ^{k\pi\mathrm{i}/n}z^{\frac{1}{2}}\right),k=0}}$	(w)^(2n) = (- 1)^(n) (lambda)^(2n) (z)^(- n)* w	(w)^(2n) == (- 1)^(n) \[Lambda]^(2n) (z)^(- n)* w	Skipped - no semantic math	Skipped - no semantic math	-	-
10.13.E11	${\displaystyle{\displaystyle\left(\vartheta^{4}-2(\nu^{2}+\mu^{2})\vartheta^{2% }+(\nu^{2}-\mu^{2})^{2}\right)w+4z^{2}(\vartheta+1)(\vartheta+2)w=0}}$ `\left(\vartheta^{4}-2(\nu^{2}+\mu^{2})\vartheta^{2}+(\nu^{2}-\mu^{2})^{2}\right)w+4z^{2}(\vartheta+1)(\vartheta+2)w = 0`	${\displaystyle{\displaystyle w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\mu% }(z).}}$	((vartheta)^(4)- 2((nu)^(2)+ (mu)^(2))(vartheta)^(2)+((nu)^(2)- (mu)^(2))^(2))w + 4(z)^(2)(vartheta + 1)(vartheta + 2)*w = 0	(\[CurlyTheta]^(4)- 2(\[Nu]^(2)+ \[Mu]^(2))\[CurlyTheta]^(2)+(\[Nu]^(2)- \[Mu]^(2))^(2))w + 4(z)^(2)(\[CurlyTheta]+ 1)(\[CurlyTheta]+ 2)*w == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
10.14#Ex1	${\displaystyle{\displaystyle\|J_{\nu}\left(x\right)\|\leq 1}}$ `\|\BesselJ{\nu}@{x}\| \leq 1`	${\displaystyle{\displaystyle\nu\geq 0,\Re(\nu+k+1)>0}}$	abs(BesselJ(nu, x)) <= 1	Abs[BesselJ[\[Nu], x]] <= 1	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.14#Ex2	${\displaystyle{\displaystyle\|J_{\nu}\left(x\right)\|\leq 2^{-\frac{1}{2}}}}$ `\|\BesselJ{\nu}@{x}\| \leq 2^{-\frac{1}{2}}`	${\displaystyle{\displaystyle\nu\geq 1,\Re(\nu+k+1)>0}}$	abs(BesselJ(nu, x)) <= (2)^(-(1)/(2))	Abs[BesselJ[\[Nu], x]] <= (2)^(-Divide[1,2])	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
10.14.E2	${\displaystyle{\displaystyle 0<J_{\nu}\left(\nu\right)}}$ `0 < \BesselJ{\nu}@{\nu}`	${\displaystyle{\displaystyle\nu>0,\Re(\nu+k+1)>0}}$	0 < BesselJ(nu, nu)	0 < BesselJ[\[Nu], \[Nu]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.14.E2	${\displaystyle{\displaystyle J_{\nu}\left(\nu\right)<\frac{2^{\frac{1}{3}}}{3^% {\frac{2}{3}}\Gamma\left(\tfrac{2}{3}\right)\nu^{\frac{1}{3}}}}}$ `\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}`	${\displaystyle{\displaystyle\nu>0,\Re(\nu+k+1)>0}}$	BesselJ(nu, nu) < ((2)^((1)/(3)))/((3)^((2)/(3))* GAMMA((2)/(3))*(nu)^((1)/(3)))	BesselJ[\[Nu], \[Nu]] < Divide[(2)^(Divide[1,3]),(3)^(Divide[2,3])* Gamma[Divide[2,3]]*\[Nu]^(Divide[1,3])]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.14.E3	${\displaystyle{\displaystyle\|J_{n}\left(z\right)\|\leq e^{\|\Im z\|}}}$ `\|\BesselJ{n}@{z}\| \leq e^{\|\imagpart@@{z}\|}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	abs(BesselJ(n, z)) <= exp(abs(Im(z)))	Abs[BesselJ[n, z]] <= Exp[Abs[Im[z]]]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.14.E4	${\displaystyle{\displaystyle\|J_{\nu}\left(z\right)\|\leq\frac{\|\tfrac{1}{2}z\|^{% \nu}e^{\|\Im z\|}}{\Gamma\left(\nu+1\right)}}}$ `\|\BesselJ{\nu}@{z}\| \leq \frac{\|\tfrac{1}{2}z\|^{\nu}e^{\|\imagpart@@{z}\|}}{\EulerGamma@{\nu+1}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}}$	abs(BesselJ(nu, z)) <= ((abs((1)/(2)z))^(nu) exp(abs(Im(z))))/(GAMMA(nu + 1))	Abs[BesselJ[\[Nu], z]] <= Divide[(Abs[Divide[1,2]z])^\[Nu] Exp[Abs[Im[z]]],Gamma[\[Nu]+ 1]]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.14.E5	${\displaystyle{\displaystyle\|J_{\nu}\left(\nu x\right)\|\leq\frac{x^{\nu}\exp% \left(\nu(1-x^{2})^{\frac{1}{2}}\right)}{\left(1+(1-x^{2})^{\frac{1}{2}}\right% )^{\nu}}}}$ `\|\BesselJ{\nu}@{\nu x}\| \leq \frac{x^{\nu}\exp@{\nu(1-x^{2})^{\frac{1}{2}}}}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}}`	${\displaystyle{\displaystyle\nu\geq 0,0<x,x\leq 1,\Re(\nu+k+1)>0}}$	abs(BesselJ(nu, nux)) <= ((x)^(nu) exp(nu*(1 - (x)^(2))^((1)/(2))))/((1 +(1 - (x)^(2))^((1)/(2)))^(nu))	Abs[BesselJ[\[Nu], \[Nu]x]] <= Divide[(x)^\[Nu] Exp[\[Nu]*(1 - (x)^(2))^(Divide[1,2])],(1 +(1 - (x)^(2))^(Divide[1,2]))^\[Nu]]	Failure	Failure	Successful [Tested: 3]	Skip - No test values generated
10.14.E7	${\displaystyle{\displaystyle 1\leq\frac{J_{\nu}\left(\nu x\right)}{x^{\nu}J_{% \nu}\left(\nu\right)}}}$ `1 \leq \frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}}`	${\displaystyle{\displaystyle\nu\geq 0,0<x,x\leq 1,\Re(\nu+k+1)>0}}$	1 <= (BesselJ(nu, nux))/((x)^(nu) BesselJ(nu, nu))	1 <= Divide[BesselJ[\[Nu], \[Nu]x],(x)^\[Nu] BesselJ[\[Nu], \[Nu]]]	Failure	Failure	Successful [Tested: 3]	Skip - No test values generated
10.14.E7	${\displaystyle{\displaystyle\frac{J_{\nu}\left(\nu x\right)}{x^{\nu}J_{\nu}% \left(\nu\right)}\leq e^{\nu(1-x)}}}$ `\frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}} \leq e^{\nu(1-x)}`	${\displaystyle{\displaystyle\nu\geq 0,0<x,x\leq 1,\Re(\nu+k+1)>0}}$	(BesselJ(nu, nux))/((x)^(nu) BesselJ(nu, nu)) <= exp(nu*(1 - x))	Divide[BesselJ[\[Nu], \[Nu]x],(x)^\[Nu] BesselJ[\[Nu], \[Nu]]] <= Exp[\[Nu]*(1 - x)]	Failure	Failure	Successful [Tested: 3]	Skip - No test values generated
10.14.E8	${\displaystyle{\displaystyle\|J_{n}\left(nz\right)\|\leq\frac{\left\|z^{n}\exp% \left(n(1-z^{2})^{\frac{1}{2}}\right)\right\|}{\left\|1+(1-z^{2})^{\frac{1}{2}}% \right\|^{n}}}}$ `\|\BesselJ{n}@{nz}\| \leq \frac{\left\|z^{n}\exp@{n(1-z^{2})^{\frac{1}{2}}}\right\|}{\left\|1+(1-z^{2})^{\frac{1}{2}}\right\|^{n}}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	abs(BesselJ(n, nz)) <= (abs((z)^(n) exp(n*(1 - (z)^(2))^((1)/(2)))))/((abs(1 +(1 - (z)^(2))^((1)/(2))))^(n))	Abs[BesselJ[n, nz]] <= Divide[Abs[(z)^(n) Exp[n*(1 - (z)^(2))^(Divide[1,2])]],(Abs[1 +(1 - (z)^(2))^(Divide[1,2])])^(n)]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.14.E9	${\displaystyle{\displaystyle\|J_{n}\left(nz\right)\|\leq 1}}$ `\|\BesselJ{n}@{nz}\| \leq 1`	${\displaystyle{\displaystyle n=0,\Re(n+k+1)>0}}$	abs(BesselJ(n, n*z)) <= 1	Abs[BesselJ[n, n*z]] <= 1	Failure	Failure	Error	Successful [Tested: 21]
10.15.E1	${\displaystyle{\displaystyle\frac{\partial J_{+\nu}\left(z\right)}{\partial\nu% }=+J_{+\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{+\nu}% \sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(k+1+\nu\right)}{\Gamma\left(k+1+\nu% \right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}}}$ `\pderiv{\BesselJ{+\nu}@{z}}{\nu} = +\BesselJ{+\nu}@{z}\ln@{\tfrac{1}{2}z}-(\tfrac{1}{2}z)^{+\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{k+1+\nu}}{\EulerGamma@{k+1+\nu}}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}`	${\displaystyle{\displaystyle\Re(k+1+\nu)>0}}$	diff(BesselJ(+ nu, z), nu) = + BesselJ(+ nu, z)ln((1)/(2)z)-((1)/(2)z)^(+ nu) sum((- 1)^(k)(Psi(k + 1 + nu))/(GAMMA(k + 1 + nu))(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)	D[BesselJ[+ \[Nu], z], \[Nu]] == + BesselJ[+ \[Nu], z]Log[Divide[1,2]z]-(Divide[1,2]z)^(+ \[Nu]) Sum[(- 1)^(k)Divide[PolyGamma[k + 1 + \[Nu]],Gamma[k + 1 + \[Nu]]]Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [7 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -2]} ... skip entries to safe data
10.15.E1	${\displaystyle{\displaystyle\frac{\partial J_{-\nu}\left(z\right)}{\partial\nu% }=-J_{-\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)+(\tfrac{1}{2}z)^{-\nu}% \sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(k+1-\nu\right)}{\Gamma\left(k+1-\nu% \right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}}}$ `\pderiv{\BesselJ{-\nu}@{z}}{\nu} = -\BesselJ{-\nu}@{z}\ln@{\tfrac{1}{2}z}+(\tfrac{1}{2}z)^{-\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{k+1-\nu}}{\EulerGamma@{k+1-\nu}}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}`	${\displaystyle{\displaystyle\Re(k+1+\nu)>0,\Re((-\nu)+k+1)>0,\Re(k+1-\nu)>0}}$	diff(BesselJ(- nu, z), nu) = - BesselJ(- nu, z)ln((1)/(2)z)+((1)/(2)z)^(- nu) sum((- 1)^(k)(Psi(k + 1 - nu))/(GAMMA(k + 1 - nu))(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)	D[BesselJ[- \[Nu], z], \[Nu]] == - BesselJ[- \[Nu], z]Log[Divide[1,2]z]+(Divide[1,2]z)^(- \[Nu]) Sum[(- 1)^(k)Divide[PolyGamma[k + 1 - \[Nu]],Gamma[k + 1 - \[Nu]]]Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [7 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.15.E2	${\displaystyle{\displaystyle\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}% =\cot\left(\nu\pi\right)\left(\frac{\partial J_{\nu}\left(z\right)}{\partial% \nu}-\pi Y_{\nu}\left(z\right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_% {-\nu}\left(z\right)}{\partial\nu}-\pi J_{\nu}\left(z\right)}}$ `\pderiv{\BesselY{\nu}@{z}}{\nu} = \cot@{\nu\pi}\left(\pderiv{\BesselJ{\nu}@{z}}{\nu}-\pi\BesselY{\nu}@{z}\right)-\csc@{\nu\pi}\pderiv{\BesselJ{-\nu}@{z}}{\nu}-\pi\BesselJ{\nu}@{z}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	diff(BesselY(nu, z), nu) = cot(nuPi)(diff(BesselJ(nu, z), nu)- PiBesselY(nu, z))- csc(nuPi)diff(BesselJ(- nu, z), nu)- PiBesselJ(nu, z)	D[BesselY[\[Nu], z], \[Nu]] == Cot[\[Nu]Pi](D[BesselJ[\[Nu], z], \[Nu]]- PiBesselY[\[Nu], z])- Csc[\[Nu]Pi]D[BesselJ[- \[Nu], z], \[Nu]]- PiBesselJ[\[Nu], z]	Successful	Failure	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.16#Ex1	${\displaystyle{\displaystyle J_{\frac{1}{2}}\left(z\right)=Y_{-\frac{1}{2}}% \left(z\right)}}$ `\BesselJ{\frac{1}{2}}@{z} = \BesselY{-\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0,\Re((-\frac{1}{2})+k+1)>0% ,\Re((-(-\frac{1}{2}))+k+1)>0}}$	BesselJ((1)/(2), z) = BesselY(-(1)/(2), z)	BesselJ[Divide[1,2], z] == BesselY[-Divide[1,2], z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
10.16#Ex1	${\displaystyle{\displaystyle Y_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z% }\right)^{\frac{1}{2}}\sin z}}$ `\BesselY{-\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sin@@{z}`	${\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0,\Re((-\frac{1}{2})+k+1)>0% ,\Re((-(-\frac{1}{2}))+k+1)>0}}$	BesselY(-(1)/(2), z) = ((2)/(Piz))^((1)/(2)) sin(z)	BesselY[-Divide[1,2], z] == (Divide[2,Piz])^(Divide[1,2]) Sin[z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.16#Ex2	${\displaystyle{\displaystyle J_{-\frac{1}{2}}\left(z\right)=-Y_{\frac{1}{2}}% \left(z\right)}}$ `\BesselJ{-\frac{1}{2}}@{z} = -\BesselY{\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0,\Re((\frac{1}{2})+k+1)>0% ,\Re((-(\frac{1}{2}))+k+1)>0}}$	BesselJ(-(1)/(2), z) = - BesselY((1)/(2), z)	BesselJ[-Divide[1,2], z] == - BesselY[Divide[1,2], z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
10.16#Ex2	${\displaystyle{\displaystyle-Y_{\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z% }\right)^{\frac{1}{2}}\cos z}}$ `-\BesselY{\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\cos@@{z}`	${\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0,\Re((\frac{1}{2})+k+1)>0% ,\Re((-(\frac{1}{2}))+k+1)>0}}$	- BesselY((1)/(2), z) = ((2)/(Piz))^((1)/(2)) cos(z)	- BesselY[Divide[1,2], z] == (Divide[2,Piz])^(Divide[1,2]) Cos[z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.16#Ex3	${\displaystyle{\displaystyle{H^{(1)}_{\frac{1}{2}}}\left(z\right)=-i{H^{(1)}_{% -\frac{1}{2}}}\left(z\right)}}$ `\HankelH{1}{\frac{1}{2}}@{z} = -i\HankelH{1}{-\frac{1}{2}}@{z}`		HankelH1((1)/(2), z) = - I*HankelH1(-(1)/(2), z)	HankelH1[Divide[1,2], z] == - I*HankelH1[-Divide[1,2], z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
10.16#Ex3	${\displaystyle{\displaystyle-i{H^{(1)}_{-\frac{1}{2}}}\left(z\right)=-i\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}e^{iz}}}$ `-i\HankelH{1}{-\frac{1}{2}}@{z} = -i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{iz}`		- IHankelH1(-(1)/(2), z) = - I((2)/(Piz))^((1)/(2)) exp(I*z)	- IHankelH1[-Divide[1,2], z] == - I(Divide[2,Piz])^(Divide[1,2]) Exp[I*z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.16#Ex4	${\displaystyle{\displaystyle{H^{(2)}_{\frac{1}{2}}}\left(z\right)=i{H^{(2)}_{-% \frac{1}{2}}}\left(z\right)}}$ `\HankelH{2}{\frac{1}{2}}@{z} = i\HankelH{2}{-\frac{1}{2}}@{z}`		HankelH2((1)/(2), z) = I*HankelH2(-(1)/(2), z)	HankelH2[Divide[1,2], z] == I*HankelH2[-Divide[1,2], z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
10.16#Ex4	${\displaystyle{\displaystyle i{H^{(2)}_{-\frac{1}{2}}}\left(z\right)=i\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-iz}}}$ `i\HankelH{2}{-\frac{1}{2}}@{z} = i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-iz}`		IHankelH2(-(1)/(2), z) = I((2)/(Piz))^((1)/(2)) exp(- I*z)	IHankelH2[-Divide[1,2], z] == I(Divide[2,Piz])^(Divide[1,2]) Exp[- I*z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.16#Ex5	${\displaystyle{\displaystyle J_{\frac{1}{4}}\left(z\right)=-2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(0,2z^{\frac{1}{2}}\right)-W% \left(0,-2z^{\frac{1}{2}}\right)\right)}}$ `\BesselJ{\frac{1}{4}}@{z} = -2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(\paraW@{0}{2z^{\frac{1}{2}}}-\paraW@{0}{-2z^{\frac{1}{2}}}\right)`	${\displaystyle{\displaystyle\Re((\frac{1}{4})+k+1)>0}}$	Error	BesselJ[Divide[1,4], z] == - (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[1,4])(Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] * Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), 2(z)^(Divide[1,2]) * Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), 2(z)^(Divide[1,2]) * Exp[Divide[PiI,4]]] )- Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] * Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), - 2(z)^(Divide[1,2]) * Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), - 2(z)^(Divide[1,2]) * Exp[Divide[Pi*I,4]]] ))	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[0.8427727646508262, -0.04212015747529019], Times[Complex[0.4703662267003617, -0.06192488852586185], Plus[Times[0.4550898605622274, Plus[Times[Complex[0.3150667711363517, -1.1318933470332309], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.1941072423227021, 0.35884759380625464], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[-0.4550898605622274, Plus[Times[Complex[1.684848183162187, 0.4798071226199044], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.8058077119758371, -1.0109338182195815], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.7942814592773979, 0.6544287188687908], Times[Complex[0.41086410074312574, -0.23721249916439713], Plus[Times[0.4550898605622274, Plus[Times[Complex[1.9382359752879499, -0.7976721648462198], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.22978077998995444, -0.1584303699393873], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[-0.4550898605622274, Plus[Times[Complex[0.8690225748967872, 1.5500248253586082], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[2.5774777701947826, 0.910783030451775], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.16#Ex6	${\displaystyle{\displaystyle J_{-\frac{1}{4}}\left(z\right)=2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(0,2z^{\frac{1}{2}}\right)+W% \left(0,-2z^{\frac{1}{2}}\right)\right)}}$ `\BesselJ{-\frac{1}{4}}@{z} = 2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(\paraW@{0}{2z^{\frac{1}{2}}}+\paraW@{0}{-2z^{\frac{1}{2}}}\right)`	${\displaystyle{\displaystyle\Re((-\frac{1}{4})+k+1)>0}}$	Error	BesselJ[-Divide[1,4], z] == (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[1,4])(Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] * Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), 2(z)^(Divide[1,2]) * Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), 2(z)^(Divide[1,2]) * Exp[Divide[PiI,4]]] )+ Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] * Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), - 2(z)^(Divide[1,2]) * Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), - 2(z)^(Divide[1,2]) * Exp[Divide[Pi*I,4]]] ))	Missing Macro Error	Aborted	-	Failed [7 / 7] Result: Plus[Complex[0.7570692040611657, -0.36205959587261455], Times[Complex[-0.4703662267003617, 0.06192488852586186], Plus[Times[0.4550898605622274, Plus[Times[Complex[0.3150667711363517, -1.1318933470332309], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.1941072423227021, 0.35884759380625464], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[1.684848183162187, 0.4798071226199044], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.8058077119758371, -1.0109338182195815], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[1.1199640481676587, -0.30003362129733535], Times[Complex[-0.41086410074312574, 0.2372124991643971], Plus[Times[0.4550898605622274, Plus[Times[Complex[1.9382359752879499, -0.7976721648462198], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.22978077998995444, -0.1584303699393873], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[0.8690225748967872, 1.5500248253586082], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[2.5774777701947826, 0.910783030451775], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.16#Ex7	${\displaystyle{\displaystyle J_{\frac{3}{4}}\left(z\right)=-2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left(0,2z^{\frac{1}{2}}\right)-W'% \left(0,-2z^{\frac{1}{2}}\right)\right)}}$ `\BesselJ{\frac{3}{4}}@{z} = -2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(\paraW'@{0}{2z^{\frac{1}{2}}}-\paraW'@{0}{-2z^{\frac{1}{2}}}\right)`	${\displaystyle{\displaystyle\Re((\frac{3}{4})+k+1)>0}}$	Error	BesselJ[Divide[3,4], z] == - (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[3,4])((D[Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] * Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), temp Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), temp Exp[Divide[PiI,4]]] ), {temp, 1}]/.temp-> 2(z)^(Divide[1,2]))- (D[Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), temp Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), temp Exp[Divide[PiI,4]]] ), {temp, 1}]/.temp-> - 2(z)^(Divide[1,2])))	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[0.5824093961234496, 0.15854248220296385], Times[Complex[0.43831154566767444, -0.18155458676026498], Plus[Times[0.4550898605622274, Plus[Times[Complex[-1.0141669743850696, 0.548925751618472], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.3595065696883391, -0.29725176260213915], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[-0.4550898605622274, Plus[Times[Complex[0.48667094453227255, 0.3574086420945919], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.16798946016445826, 1.2035861563152026], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.0836786417162193, 0.6909849218136797], Times[Complex[0.0, -0.4744249983287943], Plus[Times[-0.4550898605622274, Plus[Times[Complex[-1.52733809531001, -0.015580244977093649], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-1.3790215645615536, -1.2403191305633965], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[-0.154282678975249, -1.0920025998149403], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.302599209723706, 0.13273628577136276], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.16#Ex8	${\displaystyle{\displaystyle J_{-\frac{3}{4}}\left(z\right)=-2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left(0,2z^{\frac{1}{2}}\right)+W'% \left(0,-2z^{\frac{1}{2}}\right)\right)}}$ `\BesselJ{-\frac{3}{4}}@{z} = -2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(\paraW'@{0}{2z^{\frac{1}{2}}}+\paraW'@{0}{-2z^{\frac{1}{2}}}\right)`	${\displaystyle{\displaystyle\Re((-\frac{3}{4})+k+1)>0}}$	Error	BesselJ[-Divide[3,4], z] == - (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[3,4])((D[Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] * Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), temp Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), temp Exp[Divide[PiI,4]]] ), {temp, 1}]/.temp-> 2(z)^(Divide[1,2]))+ (D[Sqrt[(Sqrt[1+Exp[2Pi(0)]]-Exp[Pi(0)])/2] Exp[Divide[Pi(0),4]] ( Exp[I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] * ParabolicCylinderD[- 1/2 - I(0), temp Exp[-Divide[PiI,4]]] + Exp[-I(Pi/8 + Arg[GAMMA[1/2 + I(0)]]/2)] ParabolicCylinderD[- 1/2 + I(0), temp Exp[Divide[PiI,4]]] ), {temp, 1}]/.temp-> - 2(z)^(Divide[1,2])))	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[0.05605283808026881, -0.4145839244466886], Times[Complex[0.43831154566767444, -0.18155458676026498], Plus[Times[0.4550898605622274, Plus[Times[Complex[-1.0141669743850696, 0.548925751618472], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.3595065696883391, -0.29725176260213915], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[0.48667094453227255, 0.3574086420945919], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.16798946016445826, 1.2035861563152026], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.44186162583484034, -0.6708696264637843], Times[Complex[0.0, -0.4744249983287943], Plus[Times[0.4550898605622274, Plus[Times[Complex[-1.52733809531001, -0.015580244977093649], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-1.3790215645615536, -1.2403191305633965], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[-0.154282678975249, -1.0920025998149403], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.302599209723706, 0.13273628577136276], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.16.E5	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{-iz}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,+2iz\right)}}$ `\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}e^{- iz}}{\EulerGamma@{\nu+1}}\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{+ 2iz}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}}$	BesselJ(nu, z) = (((1)/(2)z)^(nu) exp(- Iz))/(GAMMA(nu + 1))KummerM(nu +(1)/(2), 2nu + 1, + 2I*z)	BesselJ[\[Nu], z] == Divide[(Divide[1,2]z)^\[Nu] Exp[- Iz],Gamma[\[Nu]+ 1]]Hypergeometric1F1[\[Nu]+Divide[1,2], 2\[Nu]+ 1, + 2I*z]	Failure	Successful	Failed [7 / 56] Result: -.827986137e-1+.7317301038I Test Values: {nu = -1/2, z = 1/23^(1/2)+1/2I} Result: -.8060140108+.3257248263I Test Values: {nu = -1/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [7 / 56] Result: Complex[-0.08279861346468581, 0.7317301035002939] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]} Result: Complex[-0.8060140105131326, 0.32572482654389856] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]} ... skip entries to safe data
10.16.E5	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{+iz}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,-2iz\right)}}$ `\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}e^{+ iz}}{\EulerGamma@{\nu+1}}\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{- 2iz}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}}$	BesselJ(nu, z) = (((1)/(2)z)^(nu) exp(+ Iz))/(GAMMA(nu + 1))KummerM(nu +(1)/(2), 2nu + 1, - 2I*z)	BesselJ[\[Nu], z] == Divide[(Divide[1,2]z)^\[Nu] Exp[+ Iz],Gamma[\[Nu]+ 1]]Hypergeometric1F1[\[Nu]+Divide[1,2], 2\[Nu]+ 1, - 2I*z]	Failure	Successful	Failed [7 / 56] Result: .827986132e-1-.7317301035I Test Values: {nu = -1/2, z = 1/23^(1/2)+1/2I} Result: .8060140102-.3257248264I Test Values: {nu = -1/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [7 / 56] Result: Complex[0.08279861346468548, -0.7317301035002935] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]} Result: Complex[0.8060140105131325, -0.325724826543898] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]} ... skip entries to safe data
10.16.E7	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{e^{-(2\nu+1)\pi i/4}}% {2^{2\nu}\Gamma\left(\nu+1\right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(+2iz\right% )}}$ `\BesselJ{\nu}@{z} = \frac{e^{-(2\nu+1)\pi i/4}}{2^{2\nu}\EulerGamma@{\nu+1}}(2z)^{-\frac{1}{2}}\WhittakerconfhyperM{0}{\nu}@{+ 2iz}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}}$	BesselJ(nu, z) = (exp(-(2nu + 1)PiI/4))/((2)^(2nu)* GAMMA(nu + 1))(2z)^(-(1)/(2))* WhittakerM(0, nu, + 2Iz)	BesselJ[\[Nu], z] == Divide[Exp[-(2\[Nu]+ 1)PiI/4],(2)^(2\[Nu])* Gamma[\[Nu]+ 1]](2z)^(-Divide[1,2])* WhittakerM[0, \[Nu], + 2Iz]	Failure	Failure	Failed [1 / 7] Result: 1.448710179-.1398527410I Test Values: {z = -1/2+1/2I*3^(1/2), nu = 1/4}	Failed [1 / 7] Result: Complex[1.448710178146189, -0.13985274040860685] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Rational[1, 4]]}
10.16.E7	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{e^{+(2\nu+1)\pi i/4}}% {2^{2\nu}\Gamma\left(\nu+1\right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(-2iz\right% )}}$ `\BesselJ{\nu}@{z} = \frac{e^{+(2\nu+1)\pi i/4}}{2^{2\nu}\EulerGamma@{\nu+1}}(2z)^{-\frac{1}{2}}\WhittakerconfhyperM{0}{\nu}@{- 2iz}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}}$	BesselJ(nu, z) = (exp(+(2nu + 1)PiI/4))/((2)^(2nu)* GAMMA(nu + 1))(2z)^(-(1)/(2))* WhittakerM(0, nu, - 2Iz)	BesselJ[\[Nu], z] == Divide[Exp[+(2\[Nu]+ 1)PiI/4],(2)^(2\[Nu])* Gamma[\[Nu]+ 1]](2z)^(-Divide[1,2])* WhittakerM[0, \[Nu], - 2Iz]	Failure	Failure	Failed [1 / 7] Result: 1.191860674-.595668984e-1I Test Values: {z = -1/23^(1/2)-1/2*I, nu = 1/4}	Failed [1 / 7] Result: Complex[1.191860673767867, -0.059566897950845576] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Rational[1, 4]]}
10.16.E9	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{\Gamma\left(\nu+1\right)}{{}_{0}F_{1}}\left(-;\nu+1;-\tfrac{1}{4}z^{2}\right% )}}$ `\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{\EulerGamma@{\nu+1}}\genhyperF{0}{1}@{-}{\nu+1}{-\tfrac{1}{4}z^{2}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}}$	BesselJ(nu, z) = (((1)/(2)z)^(nu))/(GAMMA(nu + 1))hypergeom([-], [nu + 1], -(1)/(4)*(z)^(2))	BesselJ[\[Nu], z] == Divide[(Divide[1,2]z)^\[Nu],Gamma[\[Nu]+ 1]]HypergeometricPFQ[{-}, {\[Nu]+ 1}, -Divide[1,4]*(z)^(2)]	Error	Failure	-	Error
10.17.E7	${\displaystyle{\displaystyle z^{\frac{1}{2}}=\exp\left(\tfrac{1}{2}\ln\|z\|+% \tfrac{1}{2}i\operatorname{ph}z\right)}}$ `z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{\|z\|}+\tfrac{1}{2}i\phase@@{z}}`		(z)^((1)/(2)) = exp((1)/(2)ln(abs(z))+(1)/(2)I*argument(z))	(z)^(Divide[1,2]) == Exp[Divide[1,2]Log[Abs[z]]+Divide[1,2]I*Arg[z]]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.17.E16	${\displaystyle{\displaystyle G_{p}\left(z\right)=\frac{e^{z}}{2\pi}\Gamma\left% (p\right)\Gamma\left(1-p,z\right)}}$ `\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}`	${\displaystyle{\displaystyle\Re p>0}}$	(exp(z)/(2Pi))GAMMA(p)GAMMA(1-p,z) = (exp(z))/(2Pi)GAMMA(p)GAMMA(1 - p, z)	Error	Successful	Missing Macro Error	-	-
10.17.E17	${\displaystyle{\displaystyle R_{\ell}^{+}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi% \right)\\left(\sum_{k=0}^{m-1}(+i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left% (-2iz\right)+R_{m,\ell}^{+}(\nu,z)\right)}}$ `R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)`	${\displaystyle{\displaystyle\Re(\ell-k)>0,k\geq 1}}$	(R[ell])^(+)(nu , z) = (- 1)^(ell)* 2cos(nuPi)(sum((+ I)^(k)(((4(nu)^(2)- (1)^(2))(4(nu)^(2)- (3)^(2)) .. (4(nu)^(2)-(2k - 1)^(2)))/(factorial(k)(8)^(k)))/((z)^(k))(exp(- 2Iz)/(2Pi))GAMMA(ell - k)GAMMA(1-ell - k,- 2Iz), k = 0..m - 1)+ (R[m , ell])^(+)(nu , z))	Error	Error	Missing Macro Error	-	-
10.17.E17	${\displaystyle{\displaystyle R_{\ell}^{-}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi% \right)\\left(\sum_{k=0}^{m-1}(-i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left% (+2iz\right)+R_{m,\ell}^{-}(\nu,z)\right)}}$ `R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)`	${\displaystyle{\displaystyle\Re(\ell-k)>0,k\geq 1}}$	(R[ell])^(-)(nu , z) = (- 1)^(ell)* 2cos(nuPi)(sum((- I)^(k)(((4(nu)^(2)- (1)^(2))(4(nu)^(2)- (3)^(2)) .. (4(nu)^(2)-(2k - 1)^(2)))/(factorial(k)(8)^(k)))/((z)^(k))(exp(+ 2Iz)/(2Pi))GAMMA(ell - k)GAMMA(1-ell - k,+ 2Iz), k = 0..m - 1)+ (R[m , ell])^(-)(nu , z))	Error	Error	Missing Macro Error	-	-
10.18#Ex7	${\displaystyle{\displaystyle M_{\nu}\left(x\right)=\left({J_{\nu}^{2}}\left(x% \right)+{Y_{\nu}^{2}}\left(x\right)\right)^{\frac{1}{2}}}}$ `\HankelmodM{\nu}@{x} = \left(\BesselJ{\nu}^{2}@{x}+\BesselY{\nu}^{2}@{x}\right)^{\frac{1}{2}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	Error	Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2] == ((BesselJ[\[Nu], x])^(2)+ (BesselY[\[Nu], x])^(2))^(Divide[1,2])	Missing Macro Error	Failure	-	Failed [30 / 30] Result: Complex[0.19554332981034928, -0.3390785475644471] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.7197518351343698, 1.0182547128018542] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.18#Ex8	${\displaystyle{\displaystyle N_{\nu}\left(x\right)=\left({J_{\nu}'^{2}}\left(x% \right)+{Y_{\nu}'^{2}}\left(x\right)\right)^{\frac{1}{2}}}}$ `\HankelmodderivN{\nu}@{x} = \left(\BesselJ{\nu}'^{2}@{x}+\BesselY{\nu}'^{2}@{x}\right)^{\frac{1}{2}}`		Error	Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2] == ((D[BesselJ[\[Nu], x], {x, 1}])^(2)+ (D[BesselY[\[Nu], x], {x, 1}])^(2))^(Divide[1,2])	Missing Macro Error	Failure	-	Failed [30 / 30] Result: Complex[-0.3065654786420606, 0.09106250304027241] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.41179972752410343, -0.08651542233456301] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.18.E10	${\displaystyle{\displaystyle(x^{2}-\nu^{2})M_{\nu}\left(x\right)M_{\nu}'\left(% x\right)+x^{2}N_{\nu}\left(x\right)N_{\nu}'\left(x\right)+x{N_{\nu}^{2}}\left(% x\right)=0}}$ `(x^{2}-\nu^{2})\HankelmodM{\nu}@{x}\HankelmodM{\nu}'@{x}+x^{2}\HankelmodderivN{\nu}@{x}\HankelmodderivN{\nu}'@{x}+x\HankelmodderivN{\nu}^{2}@{x} = 0`		Error	((x)^(2)- \[Nu]^(2))Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2]D[Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2], {x, 1}]+ (x)^(2)* Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2]D[Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2], {x, 1}]+ x(Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2])^(2) == 0	Missing Macro Error	Aborted	-	Failed [30 / 30] Result: Complex[0.7620133104065328, -0.7345190431210711] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.2607567755462643, -4.475082123070706] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.18.E13	${\displaystyle{\displaystyle x^{2}M_{\nu}''\left(x\right)+xM_{\nu}'\left(x% \right)+(x^{2}-\nu^{2})M_{\nu}\left(x\right)=\frac{4}{\pi^{2}{{M_{\nu}^{3}}(x)% }}}}$ `x^{2}\HankelmodM{\nu}''@{x}+x\HankelmodM{\nu}'@{x}+(x^{2}-\nu^{2})\HankelmodM{\nu}@{x} = \frac{4}{\pi^{2}{\HankelmodM{\nu}^{3}(x)}}`		Error	(x)^(2)* D[Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2], {x, 2}]+ xD[Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2], {x, 1}]+((x)^(2)- \[Nu]^(2))Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2] == Divide[4,(Pi)^(2)*(Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2])^(3)]	Missing Macro Error	Translation Error	-	-
10.20.E1	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}\zeta}{\mathrm{d}z}\right)^{% 2}=\frac{1-z^{2}}{\zeta z^{2}}}}$ `\left(\deriv{\zeta}{z}\right)^{2} = \frac{1-z^{2}}{\zeta z^{2}}`		(diff(zeta, z))^(2) = (1 - (z)^(2))/(zeta*(z)^(2))	(D[\[Zeta], z])^(2) == Divide[1 - (z)^(2),\[Zeta]*(z)^(2)]	Failure	Failure	Failed [70 / 70] Result: .8660254030+.4999999994I Test Values: {z = 1/23^(1/2)+1/2I, zeta = 1/23^(1/2)+1/2I} Result: .4999999994-.8660254030I Test Values: {z = 1/23^(1/2)+1/2I, zeta = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [70 / 70] Result: Complex[0.8660254037844386, 0.4999999999999999] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4999999999999999, -0.8660254037844386] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.20.E2	${\displaystyle{\displaystyle\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{% \sqrt{1-t^{2}}}{t}\mathrm{d}t}}$ `\frac{2}{3}\zeta^{\frac{3}{2}} = \int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t}`	${\displaystyle{\displaystyle 0<z,z\leq 1}}$	(2)/(3)*(zeta)^((3)/(2)) = int((sqrt(1 - (t)^(2)))/(t), t = z..1)	Divide[2,3]*\[Zeta]^(Divide[3,2]) == Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.20.E2	${\displaystyle{\displaystyle\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\mathrm{d}t=% \ln\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\sqrt{1-z^{2}}}}$ `\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t} = \ln@{\frac{1+\sqrt{1-z^{2}}}{z}}-\sqrt{1-z^{2}}`	${\displaystyle{\displaystyle 0<z,z\leq 1}}$	int((sqrt(1 - (t)^(2)))/(t), t = z..1) = ln((1 +sqrt(1 - (z)^(2)))/(z))-sqrt(1 - (z)^(2))	Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}, GenerateConditions->None] == Log[Divide[1 +Sqrt[1 - (z)^(2)],z]]-Sqrt[1 - (z)^(2)]	Error	Aborted	-	Skipped - Because timed out
10.20.E3	${\displaystyle{\displaystyle\frac{2}{3}(-\zeta)^{\frac{3}{2}}=\int_{1}^{z}% \frac{\sqrt{t^{2}-1}}{t}\mathrm{d}t}}$ `\frac{2}{3}(-\zeta)^{\frac{3}{2}} = \int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t}`	${\displaystyle{\displaystyle 1\leq z,z<\infty}}$	(2)/(3)*(- zeta)^((3)/(2)) = int((sqrt((t)^(2)- 1))/(t), t = 1..z)	Divide[2,3]*(- \[Zeta])^(Divide[3,2]) == Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}, GenerateConditions->None]	Failure	Aborted	Failed [20 / 20] Result: -.7483698391+.4714045210I Test Values: {z = 3/2, zeta = 1/23^(1/2)+1/2I} Result: -.2769653183-.6666666667I Test Values: {z = 3/2, zeta = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [20 / 20] Result: Complex[-0.7483698389729962, 0.4714045207910317] Test Values: {Rule[z, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.27696531818196457, -0.6666666666666666] Test Values: {Rule[z, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.20.E3	${\displaystyle{\displaystyle\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\mathrm{d}t=% \sqrt{z^{2}-1}-\operatorname{arcsec}z}}$ `\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t} = \sqrt{z^{2}-1}-\asec@@{z}`	${\displaystyle{\displaystyle 1\leq z,z<\infty}}$	int((sqrt((t)^(2)- 1))/(t), t = 1..z) = sqrt((z)^(2)- 1)- arcsec(z)	Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}, GenerateConditions->None] == Sqrt[(z)^(2)- 1]- ArcSec[z]	Failure	Aborted	Successful [Tested: 2]	Successful [Tested: 2]
10.20#Ex1	${\displaystyle{\displaystyle A_{0}(0)=1}}$ `A_{0}(0) = 1`		A[0](0) = 1	Subscript[A, 0][0] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex2	${\displaystyle{\displaystyle A_{1}(0)=-\tfrac{1}{225}}}$ `A_{1}(0) = -\tfrac{1}{225}`		A[1](0) = -(1)/(225)	Subscript[A, 1][0] == -Divide[1,225]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex3	${\displaystyle{\displaystyle A_{2}(0)=\tfrac{1\;51439}{2182\;95000}}}$ `A_{2}(0) = \tfrac{1\;51439}{2182\;95000}`		A[2](0) = (151439)/(218295000)	Subscript[A, 2][0] == Divide[151439,218295000]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex4	${\displaystyle{\displaystyle A_{3}(0)=-\tfrac{8872\;78009}{250\;49351\;25000}}}$ `A_{3}(0) = -\tfrac{8872\;78009}{250\;49351\;25000}`		A[3](0) = -(887278009)/(2504935125000)	Subscript[A, 3][0] == -Divide[887278009,2504935125000]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex5	${\displaystyle{\displaystyle B_{0}(0)=\tfrac{1}{70}2^{\frac{1}{3}}}}$ `B_{0}(0) = \tfrac{1}{70}2^{\frac{1}{3}}`		B[0](0) = (1)/(70)*(2)^((1)/(3))	Subscript[B, 0][0] == Divide[1,70]*(2)^(Divide[1,3])	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex6	${\displaystyle{\displaystyle B_{1}(0)=-\tfrac{1213}{10\;23750}2^{\frac{1}{3}}}}$ `B_{1}(0) = -\tfrac{1213}{10\;23750}2^{\frac{1}{3}}`		B[1](0) = -(1213)/(1023750)*(2)^((1)/(3))	Subscript[B, 1][0] == -Divide[1213,1023750]*(2)^(Divide[1,3])	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex7	${\displaystyle{\displaystyle B_{2}(0)=\tfrac{1\;65425\;37833}{3774\;32055\;000% 00}2^{\frac{1}{3}}}}$ `B_{2}(0) = \tfrac{1\;65425\;37833}{3774\;32055\;00000}2^{\frac{1}{3}}`		B[2](0) = (16542537833)/(37743205500000)*(2)^((1)/(3))	Subscript[B, 2][0] == Divide[16542537833,37743205500000]*(2)^(Divide[1,3])	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20#Ex8	${\displaystyle{\displaystyle B_{3}(0)=-\tfrac{959\;71711\;84603}{25\;47666\;37% 125\;00000}2^{\frac{1}{3}}}}$ `B_{3}(0) = -\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3}}`		B[3](0) = -(9597171184603)/(25476663712500000)*(2)^((1)/(3))	Subscript[B, 3][0] == -Divide[9597171184603,25476663712500000]*(2)^(Divide[1,3])	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20.E15	${\displaystyle{\displaystyle\zeta=(\tfrac{3}{2})^{\frac{2}{3}}(\tau-i\pi)^{% \frac{2}{3}}}}$ `\zeta = (\tfrac{3}{2})^{\frac{2}{3}}(\tau- i\pi)^{\frac{2}{3}}`	${\displaystyle{\displaystyle 0\leq\tau,\tau<\infty}}$	zeta = ((3)/(2))^((2)/(3))(tau - IPi)^((2)/(3))	\[Zeta] == (Divide[3,2])^(Divide[2,3])(\[Tau]- IPi)^(Divide[2,3])	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20.E16	${\displaystyle{\displaystyle\zeta=e^{-i\pi/3}\tau}}$ `\zeta = e^{- i\pi/3}\tau`	${\displaystyle{\displaystyle 0\leq\tau,\tau\leq(\tfrac{3}{2}\pi)^{\frac{2}{3}}}}$	zeta = exp(- IPi/3)tau	\[Zeta] == Exp[- IPi/3]\[Tau]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.20.E17	${\displaystyle{\displaystyle z=+(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}+\mathrm% {i}(\tau^{2}-\tau\tanh\tau)^{\frac{1}{2}}}}$ `z = +(\tau\coth@@{\tau}-\tau^{2})^{\frac{1}{2}}+\iunit(\tau^{2}-\tau\tanh@@{\tau})^{\frac{1}{2}}`	${\displaystyle{\displaystyle 0\leq\tau,\tau\leq\tau_{0}}}$	z = +(taucoth(tau)- (tau)^(2))^((1)/(2))+ I((tau)^(2)- tau*tanh(tau))^((1)/(2))	z == +(\[Tau]Coth[\[Tau]]- \[Tau]^(2))^(Divide[1,2])+ I(\[Tau]^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2])	Failure	Failure	Failed [21 / 21] Result: .8660254040-1.214547924I Test Values: {tau = 3/2, z = 1/23^(1/2)+1/2I} Result: -.5000000000-.8485225201I Test Values: {tau = 3/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Skip - No test values generated
10.20.E17	${\displaystyle{\displaystyle z=-(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}-\mathrm% {i}(\tau^{2}-\tau\tanh\tau)^{\frac{1}{2}}}}$ `z = -(\tau\coth@@{\tau}-\tau^{2})^{\frac{1}{2}}-\iunit(\tau^{2}-\tau\tanh@@{\tau})^{\frac{1}{2}}`	${\displaystyle{\displaystyle 0\leq\tau,\tau\leq\tau_{0}}}$	z = -(taucoth(tau)- (tau)^(2))^((1)/(2))- I((tau)^(2)- tau*tanh(tau))^((1)/(2))	z == -(\[Tau]Coth[\[Tau]]- \[Tau]^(2))^(Divide[1,2])- I(\[Tau]^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2])	Failure	Failure	Failed [21 / 21] Result: .8660254040+2.214547924I Test Values: {tau = 3/2, z = 1/23^(1/2)+1/2I} Result: -.5000000000+2.580573328I Test Values: {tau = 3/2, z = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Skip - No test values generated
10.21#Ex5	${\displaystyle{\displaystyle\rho_{\nu}(0)=0}}$ `\rho_{\nu}(0) = 0`		rho[nu](0) = 0	Subscript[\[Rho], \[Nu]][0] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21.E11	${\displaystyle{\displaystyle 2\rho_{\nu}^{2}\frac{\mathrm{d}\rho_{\nu}}{% \mathrm{d}t}\frac{{\mathrm{d}}^{3}\rho_{\nu}}{{\mathrm{d}t}^{3}}-3\rho_{\nu}^{% 2}\\left(\frac{{\mathrm{d}}^{2}\rho_{\nu}}{{\mathrm{d}t}^{2}}\right)^{2}-4\pi% ^{2}\rho_{\nu}^{2}\\left(\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{2}+% (4\rho_{\nu}^{2}+1-4\nu^{2})\left(\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}% \right)^{4}=0}}$ `2\rho_{\nu}^{2}\deriv{\rho_{\nu}}{t}\deriv[3]{\rho_{\nu}}{t}-3\rho_{\nu}^{2}\\left(\deriv[2]{\rho_{\nu}}{t}\right)^{2}-4\pi^{2}\rho_{\nu}^{2}\\left(\deriv{\rho_{\nu}}{t}\right)^{2}+(4\rho_{\nu}^{2}+1-4\nu^{2})\left(\deriv{\rho_{\nu}}{t}\right)^{4} = 0`		2(rho[nu])^(2)diff(rho[nu], t)diff(rho[nu], [t$(3)])- 3(rho[nu])^(2)(diff(rho[nu], [t$(2)]))^(2)- 4(Pi)^(2)* (rho[nu])^(2)(diff(rho[nu], t))^(2)+(4(rho[nu])^(2)+ 1 - 4(nu)^(2))(diff(rho[nu], t))^(4) = 0	2(Subscript[\[Rho], \[Nu]])^(2)D[Subscript[\[Rho], \[Nu]], t]D[Subscript[\[Rho], \[Nu]], {t, 3}]- 3(Subscript[\[Rho], \[Nu]])^(2)(D[Subscript[\[Rho], \[Nu]], {t, 2}])^(2)- 4(Pi)^(2)* (Subscript[\[Rho], \[Nu]])^(2)(D[Subscript[\[Rho], \[Nu]], t])^(2)+(4(Subscript[\[Rho], \[Nu]])^(2)+ 1 - 4\[Nu]^(2))(D[Subscript[\[Rho], \[Nu]], t])^(4) == 0	Successful	Successful	-	Successful [Tested: 300]
10.21.E17	${\displaystyle{\displaystyle\frac{\mathrm{d}c}{\mathrm{d}\nu}=2c\int_{0}^{% \infty}K_{0}\left(2c\sinh t\right)e^{-2\nu t}\mathrm{d}t}}$ `\deriv{c}{\nu} = 2c\int_{0}^{\infty}\modBesselK{0}@{2c\sinh@@{t}}e^{-2\nu t}\diff{t}`		diff(c, nu) = 2cint(BesselK(0, 2csinh(t))exp(- 2nu*t), t = 0..infinity)	D[c, \[Nu]] == 2cIntegrate[BesselK[0, 2cSinh[t]]Exp[- 2\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.21#Ex19	${\displaystyle{\displaystyle\alpha_{0}=1}}$ `\alpha_{0} = 1`		alpha[0] = 1	Subscript[\[Alpha], 0] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex20	${\displaystyle{\displaystyle\alpha_{1}=\alpha}}$ `\alpha_{1} = \alpha`		alpha[1] = alpha	Subscript[\[Alpha], 1] == \[Alpha]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex21	${\displaystyle{\displaystyle\alpha_{2}=\tfrac{3}{10}\alpha^{2}}}$ `\alpha_{2} = \tfrac{3}{10}\alpha^{2}`		alpha[2] = (3)/(10)*(alpha)^(2)	Subscript[\[Alpha], 2] == Divide[3,10]*\[Alpha]^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex22	${\displaystyle{\displaystyle\alpha_{3}=-\tfrac{1}{350}\alpha^{3}+\tfrac{1}{70}}}$ `\alpha_{3} = -\tfrac{1}{350}\alpha^{3}+\tfrac{1}{70}`		alpha[3] = -(1)/(350)*(alpha)^(3)+(1)/(70)	Subscript[\[Alpha], 3] == -Divide[1,350]*\[Alpha]^(3)+Divide[1,70]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex23	${\displaystyle{\displaystyle\alpha_{4}=-\tfrac{479}{63000}\alpha^{4}-\tfrac{1}% {3150}\alpha}}$ `\alpha_{4} = -\tfrac{479}{63000}\alpha^{4}-\tfrac{1}{3150}\alpha`		alpha[4] = -(479)/(63000)(alpha)^(4)-(1)/(3150)alpha	Subscript[\[Alpha], 4] == -Divide[479,63000]\[Alpha]^(4)-Divide[1,3150]\[Alpha]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex24	${\displaystyle{\displaystyle\alpha_{5}=\tfrac{20231}{80\;85000}\alpha^{5}-% \tfrac{551}{1\;61700}\alpha^{2}}}$ `\alpha_{5} = \tfrac{20231}{80\;85000}\alpha^{5}-\tfrac{551}{1\;61700}\alpha^{2}`		alpha[5] = (20231)/(8085000)(alpha)^(5)-(551)/(161700)(alpha)^(2)	Subscript[\[Alpha], 5] == Divide[20231,8085000]\[Alpha]^(5)-Divide[551,161700]\[Alpha]^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21.E46	${\displaystyle{\displaystyle a=\tfrac{1}{2}\ln 3}}$ `a = \tfrac{1}{2}\ln@@{3}`		a = (1)/(2)*ln(3)	a == Divide[1,2]*Log[3]	Failure	Failure	Failed [6 / 6] Result: -2.049306144 Test Values: {a = -3/2} Result: .9506938555 Test Values: {a = 3/2} ... skip entries to safe data	Failed [6 / 6] Result: -2.049306144334055 Test Values: {Rule[a, -1.5]} Result: 0.9506938556659451 Test Values: {Rule[a, 1.5]} ... skip entries to safe data
10.21.E46	${\displaystyle{\displaystyle\tfrac{1}{2}\ln 3=0.54931\ldots}}$ `\tfrac{1}{2}\ln@@{3} = 0.54931\dotsc`		(1)/(2)*ln(3) = 0.54931 ..	Divide[1,2]*Log[3] == 0.54931 \[Ellipsis]	Error	Failure	Skip - symbolical successful subtest	Failed [1 / 1] Result: Plus[0.5493061443340549, Times[-0.54931, …]] Test Values: {}
10.21#Ex51	${\displaystyle{\displaystyle\alpha=\frac{(m-1)\pi}{\lambda-1}}}$ `\alpha = \frac{(m-1)\pi}{\lambda-1}`		alpha = ((m - 1)*Pi)/(lambda - 1)	\[Alpha] == Divide[(m - 1)*Pi,\[Lambda]- 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex52	${\displaystyle{\displaystyle p=\frac{\mu+3}{8\lambda}}}$ `p = \frac{\mu+3}{8\lambda}`		p = (mu + 3)/(8*lambda)	p == Divide[\[Mu]+ 3,8*\[Lambda]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex53	${\displaystyle{\displaystyle q=\frac{(\mu^{2}+46\mu-63)(\lambda^{3}-1)}{6(4% \lambda)^{3}(\lambda-1)}}}$ `q = \frac{(\mu^{2}+46\mu-63)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda-1)}`		q = (((mu)^(2)+ 46mu - 63)((lambda)^(3)- 1))/(6(4lambda)^(3)*(lambda - 1))	q == Divide[(\[Mu]^(2)+ 46\[Mu]- 63)(\[Lambda]^(3)- 1),6(4\[Lambda])^(3)*(\[Lambda]- 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex54	${\displaystyle{\displaystyle r=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)(\lambda% ^{5}-1)}{5(4\lambda)^{5}(\lambda-1)}}}$ `r = \frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)(\lambda^{5}-1)}{5(4\lambda)^{5}(\lambda-1)}`		(((mu - 1)((mu)^(2)- 114mu + 1073)((lambda)^(5)- 1))/(5(4lambda)^(5)(lambda - 1))) = (((mu)^(3)+ 185(mu)^(2)- 2053mu + 1899)((lambda)^(5)- 1))/(5(4lambda)^(5)(lambda - 1))	(Divide[(\[Mu]- 1)(\[Mu]^(2)- 114\[Mu]+ 1073)(\[Lambda]^(5)- 1),5(4\[Lambda])^(5)(\[Lambda]- 1)]) == Divide[(\[Mu]^(3)+ 185\[Mu]^(2)- 2053\[Mu]+ 1899)(\[Lambda]^(5)- 1),5(4\[Lambda])^(5)(\[Lambda]- 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex55	${\displaystyle{\displaystyle\alpha=\frac{(m-\tfrac{1}{2})\pi}{\lambda-1}}}$ `\alpha = \frac{(m-\tfrac{1}{2})\pi}{\lambda-1}`		alpha = ((m -(1)/(2))*Pi)/(lambda - 1)	\[Alpha] == Divide[(m -Divide[1,2])*Pi,\[Lambda]- 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex56	${\displaystyle{\displaystyle p=\frac{(\mu+3)\lambda-(\mu-1)}{8\lambda(\lambda-% 1)}}}$ `p = \frac{(\mu+3)\lambda-(\mu-1)}{8\lambda(\lambda-1)}`		p = ((mu + 3)lambda -(mu - 1))/(8lambda*(lambda - 1))	p == Divide[(\[Mu]+ 3)\[Lambda]-(\[Mu]- 1),8\[Lambda]*(\[Lambda]- 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex57	${\displaystyle{\displaystyle q=\frac{(\mu^{2}+46\mu-63)\lambda^{3}-(\mu-1)(\mu% -25)}{6(4\lambda)^{3}(\lambda-1)}}}$ `q = \frac{(\mu^{2}+46\mu-63)\lambda^{3}-(\mu-1)(\mu-25)}{6(4\lambda)^{3}(\lambda-1)}`		q = (((mu)^(2)+ 46mu - 63)(lambda)^(3)-(mu - 1)(mu - 25))/(6(4lambda)^(3)(lambda - 1))	q == Divide[(\[Mu]^(2)+ 46\[Mu]- 63)\[Lambda]^(3)-(\[Mu]- 1)(\[Mu]- 25),6(4\[Lambda])^(3)(\[Lambda]- 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.21#Ex58	${\displaystyle{\displaystyle r=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^% {5}-(\mu-1)(\mu^{2}-114\mu+1073)}{5(4\lambda)^{5}(\lambda-1)}}}$ `r = \frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^{5}-(\mu-1)(\mu^{2}-114\mu+1073)}{5(4\lambda)^{5}(\lambda-1)}`		(((mu - 1)((mu)^(2)- 114mu + 1073)((lambda)^(5)- 1))/(5(4lambda)^(5)(lambda - 1))) = (((mu)^(3)+ 185(mu)^(2)- 2053mu + 1899)(lambda)^(5)-(mu - 1)((mu)^(2)- 114mu + 1073))/(5(4lambda)^(5)(lambda - 1))	(Divide[(\[Mu]- 1)(\[Mu]^(2)- 114\[Mu]+ 1073)(\[Lambda]^(5)- 1),5(4\[Lambda])^(5)(\[Lambda]- 1)]) == Divide[(\[Mu]^(3)+ 185\[Mu]^(2)- 2053\[Mu]+ 1899)\[Lambda]^(5)-(\[Mu]- 1)(\[Mu]^(2)- 114\[Mu]+ 1073),5(4\[Lambda])^(5)(\[Lambda]- 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.22.E8	${\displaystyle{\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)\mathrm{d}t=2\sum% _{k=0}^{\infty}J_{\nu+2k+1}\left(x\right)}}$ `\int_{0}^{x}\BesselJ{\nu}@{t}\diff{t} = 2\sum_{k=0}^{\infty}\BesselJ{\nu+2k+1}@{x}`	${\displaystyle{\displaystyle\Re\nu>-1,\Re(\nu+k+1)>0,\Re((\nu+2k+1)+k+1)>0}}$	int(BesselJ(nu, t), t = 0..x) = 2sum(BesselJ(nu + 2k + 1, x), k = 0..infinity)	Integrate[BesselJ[\[Nu], t], {t, 0, x}, GenerateConditions->None] == 2Sum[BesselJ[\[Nu]+ 2k + 1, x], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [2 / 24] Result: -.277492396 Test Values: {nu = -1/2, x = 3/2} Result: -.1653166018 Test Values: {nu = 1/2, x = 3/2}	Skipped - Because timed out
10.22.E9	${\displaystyle{\displaystyle\int_{0}^{x}J_{2n}\left(t\right)\mathrm{d}t=\int_{% 0}^{x}J_{0}\left(t\right)\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),% \quad\int_{0}^{x}J_{2n+1}\left(t\right)\mathrm{d}t}}$ `\int_{0}^{x}\BesselJ{2n}@{t}\diff{t} = \int_{0}^{x}\BesselJ{0}@{t}\diff{t}-2\sum_{k=0}^{n-1}\BesselJ{2k+1}@{x},\quad\int_{0}^{x}\BesselJ{2n+1}@{t}\diff{t}`	${\displaystyle{\displaystyle\Re((2n)+k+1)>0,\Re(0+k+1)>0,\Re((2k+1)+k+1)>0,\Re% ((2n+1)+k+1)>0}}$	int(BesselJ(2n, t), t = 0..x) = int(BesselJ(0, t), t = 0..x)- 2sum(BesselJ(2*k + 1, x), k = 0..n - 1)	Integrate[BesselJ[2n, t], {t, 0, x}, GenerateConditions->None] == Integrate[BesselJ[0, t], {t, 0, x}, GenerateConditions->None]- 2Sum[BesselJ[2*k + 1, x], {k, 0, n - 1}, GenerateConditions->None]	Failure	Failure	Error	Error
10.22.E9	${\displaystyle{\displaystyle\int_{0}^{x}J_{0}\left(t\right)\mathrm{d}t-2\sum_{% k=0}^{n-1}J_{2k+1}\left(x\right),\quad\int_{0}^{x}J_{2n+1}\left(t\right)% \mathrm{d}t=1-J_{0}\left(x\right)-2\sum_{k=1}^{n}J_{2k}\left(x\right)}}$ `\int_{0}^{x}\BesselJ{0}@{t}\diff{t}-2\sum_{k=0}^{n-1}\BesselJ{2k+1}@{x},\quad\int_{0}^{x}\BesselJ{2n+1}@{t}\diff{t} = 1-\BesselJ{0}@{x}-2\sum_{k=1}^{n}\BesselJ{2k}@{x}`	${\displaystyle{\displaystyle\Re((2n)+k+1)>0,\Re(0+k+1)>0,\Re((2k+1)+k+1)>0,\Re% ((2n+1)+k+1)>0,\Re((2k)+k+1)>0}}$	int(BesselJ(0, t), t = 0..x)- 2sum(BesselJ(2k + 1, x), k = 0..n - 1)	Integrate[BesselJ[0, t], {t, 0, x}, GenerateConditions->None]- 2Sum[BesselJ[2k + 1, x], {k, 0, n - 1}, GenerateConditions->None]	Failure	Failure	Error	Error
10.22.E10	${\displaystyle{\displaystyle\int_{0}^{x}t^{\mu}J_{\nu}\left(t\right)\mathrm{d}% t=x^{\mu}\frac{\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{% \Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}\\sum_{k=0}^{% \infty}\frac{(\nu+2k+1)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k% \right)}{\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}J_{\nu% +2k+1}\left(x\right)}}$ `\int_{0}^{x}t^{\mu}\BesselJ{\nu}@{t}\diff{t} = x^{\mu}\frac{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}}\\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k}}{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k}}\BesselJ{\nu+2k+1}@{x}`	${\displaystyle{\displaystyle\Re\left(\mu+\nu+1\right)>0,\Re(\nu+k+1)>0,\Re((% \nu+2k+1)+k+1)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}% {2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1% }{2}+k)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k)>0}}$	int((t)^(mu)* BesselJ(nu, t), t = 0..x) = (x)^(mu)(GAMMA((1)/(2)nu +(1)/(2)mu +(1)/(2)))/(GAMMA((1)/(2)nu -(1)/(2)mu +(1)/(2))) sum(((nu + 2k + 1)GAMMA((1)/(2)nu -(1)/(2)mu +(1)/(2)+ k))/(GAMMA((1)/(2)nu +(1)/(2)mu +(3)/(2)+ k))BesselJ(nu + 2k + 1, x), k = 0..infinity)	Integrate[(t)^\[Mu]* BesselJ[\[Nu], t], {t, 0, x}, GenerateConditions->None] == (x)^\[Mu]Divide[Gamma[Divide[1,2]\[Nu]+Divide[1,2]\[Mu]+Divide[1,2]],Gamma[Divide[1,2]\[Nu]-Divide[1,2]\[Mu]+Divide[1,2]]] Sum[Divide[(\[Nu]+ 2k + 1)Gamma[Divide[1,2]\[Nu]-Divide[1,2]\[Mu]+Divide[1,2]+ k],Gamma[Divide[1,2]\[Nu]+Divide[1,2]\[Mu]+Divide[3,2]+ k]]BesselJ[\[Nu]+ 2k + 1, x], {k, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Skipped - Because timed out
10.22.E11	${\displaystyle{\displaystyle\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\mathrm% {d}t=\frac{1}{2}\sum_{k=1}^{\infty}\frac{\psi\left(k+1\right)-\psi\left(1% \right)}{k!}(\tfrac{1}{2}x)^{k}J_{k}\left(x\right)}}$ `\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = \frac{1}{2}\sum_{k=1}^{\infty}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\BesselJ{k}@{x}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}}$	int((1 - BesselJ(0, t))/(t), t = 0..x) = (1)/(2)sum((Psi(k + 1)- Psi(1))/(factorial(k))((1)/(2)x)^(k) BesselJ(k, x), k = 1..infinity)	Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}, GenerateConditions->None] == Divide[1,2]Sum[Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!](Divide[1,2]x)^(k) BesselJ[k, x], {k, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Successful [Tested: 3]	Failed [3 / 3] Result: Plus[0.2622772441151432, Times[-0.5, NSum[Times[Power[0.75, k], BesselJ[k, 1.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]} Result: Plus[0.03100698635091531, Times[-0.5, NSum[Times[Power[0.25, k], BesselJ[k, 0.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]} ... skip entries to safe data
10.22.E12	${\displaystyle{\displaystyle x\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}% \mathrm{d}t=2\sum_{k=0}^{\infty}(2k+3)(\psi\left(k+2\right)-\psi\left(1\right)% )J_{2k+3}\left(x\right)}}$ `x\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = 2\sum_{k=0}^{\infty}(2k+3)(\digamma@{k+2}-\digamma@{1})\BesselJ{2k+3}@{x}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k+3)+k+1)>0}}$	xint((1 - BesselJ(0, t))/(t), t = 0..x) = 2sum((2k + 3)(Psi(k + 2)- Psi(1))BesselJ(2k + 3, x), k = 0..infinity)	xIntegrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}, GenerateConditions->None] == 2Sum[(2k + 3)(PolyGamma[k + 2]- PolyGamma[1])BesselJ[2k + 3, x], {k, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 3]	Skipped - Because timed out
10.22.E12	${\displaystyle{\displaystyle 2\sum_{k=0}^{\infty}(2k+3)(\psi\left(k+2\right)-% \psi\left(1\right))J_{2k+3}\left(x\right)=x-2J_{1}\left(x\right)+2\sum_{k=0}^{% \infty}(2k+5)\(\psi\left(k+3\right)-\psi\left(1\right)-1)J_{2k+5}\left(x% \right)}}$ `2\sum_{k=0}^{\infty}(2k+3)(\digamma@{k+2}-\digamma@{1})\BesselJ{2k+3}@{x} = x-2\BesselJ{1}@{x}+2\sum_{k=0}^{\infty}(2k+5)\(\digamma@{k+3}-\digamma@{1}-1)\BesselJ{2k+5}@{x}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k+3)+k+1)>0,\Re(1+k+1)>0,\Re((2% k+5)+k+1)>0}}$	2sum((2k + 3)(Psi(k + 2)- Psi(1))BesselJ(2k + 3, x), k = 0..infinity) = x - 2BesselJ(1, x)+ 2sum((2k + 5)(Psi(k + 3)- Psi(1)- 1)BesselJ(2*k + 5, x), k = 0..infinity)	2Sum[(2k + 3)(PolyGamma[k + 2]- PolyGamma[1])BesselJ[2k + 3, x], {k, 0, Infinity}, GenerateConditions->None] == x - 2BesselJ[1, x]+ 2Sum[(2k + 5)(PolyGamma[k + 3]- PolyGamma[1]- 1)BesselJ[2*k + 5, x], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Aborted	Successful [Tested: 3]	Skipped - Because timed out
10.22.E13	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{2\nu}\left(2z\cos% \theta\right)\cos\left(2\mu\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi J_{% \nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}`	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re((2\nu)+k+1)>0,\Re((\nu+% \mu)+k+1)>0,\Re((\nu-\mu)+k+1)>0}}$	int(BesselJ(2nu, 2zcos(theta))cos(2mutheta), theta = 0..(1)/(2)Pi) = (1)/(2)PiBesselJ(nu + mu, z)BesselJ(nu - mu, z)	Integrate[BesselJ[2\[Nu], 2zCos[\[Theta]]]Cos[2\[Mu]\[Theta]], {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[1,2]PiBesselJ[\[Nu]+ \[Mu], z]BesselJ[\[Nu]- \[Mu], z]	Failure	Failure	Manual Skip!	Skipped - Because timed out
10.22.E14	${\displaystyle{\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)% \cos\left(2\mu\theta\right)\mathrm{d}\theta=\pi\cos\left(\mu\pi\right)J_{\nu+% \mu}\left(z\right)J_{\nu-\mu}\left(z\right)}}$ `\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \pi\cos@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}`	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re((2\nu)+k+1)>0,\Re((\nu+% \mu)+k+1)>0,\Re((\nu-\mu)+k+1)>0}}$	int(BesselJ(2nu, 2zsin(theta))cos(2mutheta), theta = 0..Pi) = Picos(muPi)BesselJ(nu + mu, z)BesselJ(nu - mu, z)	Integrate[BesselJ[2\[Nu], 2zSin[\[Theta]]]Cos[2\[Mu]\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == PiCos[\[Mu]Pi]BesselJ[\[Nu]+ \[Mu], z]BesselJ[\[Nu]- \[Mu], z]	Failure	Failure	Manual Skip!	Skipped - Because timed out
10.22.E15	${\displaystyle{\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)% \sin\left(2\mu\theta\right)\mathrm{d}\theta=\pi\sin\left(\mu\pi\right)J_{\nu+% \mu}\left(z\right)J_{\nu-\mu}\left(z\right)}}$ `\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\sin@{2\mu\theta}\diff{\theta} = \pi\sin@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}`	${\displaystyle{\displaystyle\Re\nu>-1,\Re((2\nu)+k+1)>0,\Re((\nu+\mu)+k+1)>0,% \Re((\nu-\mu)+k+1)>0}}$	int(BesselJ(2nu, 2zsin(theta))sin(2mutheta), theta = 0..Pi) = Pisin(muPi)BesselJ(nu + mu, z)BesselJ(nu - mu, z)	Integrate[BesselJ[2\[Nu], 2zSin[\[Theta]]]Sin[2\[Mu]\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == PiSin[\[Mu]Pi]BesselJ[\[Nu]+ \[Mu], z]BesselJ[\[Nu]- \[Mu], z]	Failure	Failure	Manual Skip!	Skipped - Because timed out
10.22.E16	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta% \right)\cos\left(2n\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi{J_{n}^{2}}% \left(z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}^{2}@{z}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(n+k+1)>0}}$	int(BesselJ(0, 2zsin(theta))cos(2ntheta), theta = 0..(1)/(2)Pi) = (1)/(2)Pi(BesselJ(n, z))^(2)	Integrate[BesselJ[0, 2zSin[\[Theta]]]Cos[2n\[Theta]], {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[1,2]Pi(BesselJ[n, z])^(2)	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
10.22.E17	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos% \theta\right)\cos\left(2\mu\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi\cot% \left(2\nu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right)-\tfrac{% 1}{2}\pi\csc\left(2\nu\pi\right)J_{\mu-\nu}\left(z\right)J_{-\mu-\nu}\left(z% \right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselY{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\cot@{2\nu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}-\tfrac{1}{2}\pi\csc@{2\nu\pi}\BesselJ{\mu-\nu}@{z}\BesselJ{-\mu-\nu}@{z}`	${\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\nu,\Re\nu<\tfrac{1}{2},\Re((\nu+% \mu)+k+1)>0,\Re((\nu-\mu)+k+1)>0,\Re((\mu-\nu)+k+1)>0,\Re((-\mu-\nu)+k+1)>0,% \Re((2\nu)+k+1)>0,\Re((-(2\nu))+k+1)>0}}$	int(BesselY(2nu, 2zcos(theta))cos(2mutheta), theta = 0..(1)/(2)Pi) = (1)/(2)Picot(2nuPi)BesselJ(nu + mu, z)BesselJ(nu - mu, z)-(1)/(2)Picsc(2nuPi)BesselJ(mu - nu, z)*BesselJ(- mu - nu, z)	Integrate[BesselY[2\[Nu], 2zCos[\[Theta]]]Cos[2\[Mu]\[Theta]], {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[1,2]PiCot[2\[Nu]Pi]BesselJ[\[Nu]+ \[Mu], z]BesselJ[\[Nu]- \[Mu], z]-Divide[1,2]PiCsc[2\[Nu]Pi]BesselJ[\[Mu]- \[Nu], z]*BesselJ[- \[Mu]- \[Nu], z]	Failure	Failure	Error	Skip - No test values generated
10.22.E18	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{0}\left(2z\sin\theta% \right)\cos\left(2n\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi J_{n}\left(z% \right)Y_{n}\left(z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselY{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}@{z}\BesselY{n}@{z}`	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(0+k+1)>0,\Re((-0)+k+1)>0,\Re((-n)% +k+1)>0}}$	int(BesselY(0, 2zsin(theta))cos(2ntheta), theta = 0..(1)/(2)Pi) = (1)/(2)PiBesselJ(n, z)*BesselY(n, z)	Integrate[BesselY[0, 2zSin[\[Theta]]]Cos[2n\[Theta]], {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[1,2]PiBesselJ[n, z]*BesselY[n, z]	Failure	Failure	Successful [Tested: 7]	Skipped - Because timed out
10.22.E19	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)(\sin\theta)^{\mu+1}(\cos\theta)^{2\nu+1}\mathrm{d}\theta=2^{\nu}\Gamma% \left(\nu+1\right)z^{-\nu-1}J_{\mu+\nu+1}\left(z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu+1}(\cos@@{\theta})^{2\nu+1}\diff{\theta} = 2^{\nu}\EulerGamma@{\nu+1}z^{-\nu-1}\BesselJ{\mu+\nu+1}@{z}`	${\displaystyle{\displaystyle\Re\mu>-1,\Re\nu>-1,\Re((\mu)+k+1)>0,\Re((\mu+\nu+% 1)+k+1)>0,\Re(\nu+1)>0}}$	int(BesselJ(mu, zsin(theta))(sin(theta))^(mu + 1)(cos(theta))^(2nu + 1), theta = 0..(1)/(2)Pi) = (2)^(nu) GAMMA(nu + 1)(z)^(- nu - 1) BesselJ(mu + nu + 1, z)	Integrate[BesselJ[\[Mu], zSin[\[Theta]]](Sin[\[Theta]])^(\[Mu]+ 1)(Cos[\[Theta]])^(2\[Nu]+ 1), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == (2)^\[Nu] Gamma[\[Nu]+ 1](z)^(- \[Nu]- 1) BesselJ[\[Mu]+ \[Nu]+ 1, z]	Successful	Aborted	-	Successful [Tested: 300]
10.22.E20	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)(\sin\theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta=\pi^{\frac{1}{2}}% 2^{\mu-1}z^{-\mu}\\Gamma\left(\mu+\tfrac{1}{2}\right){J_{\mu}^{2}}\left(% \tfrac{1}{2}z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu}(\cos@@{\theta})^{2\mu}\diff{\theta} = \pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\\EulerGamma@{\mu+\tfrac{1}{2}}\BesselJ{\mu}^{2}@{\tfrac{1}{2}z}`	${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re((\mu)+k+1)>0,\Re(\mu+% \tfrac{1}{2})>0}}$	int(BesselJ(mu, zsin(theta))(sin(theta))^(mu)(cos(theta))^(2mu), theta = 0..(1)/(2)Pi) = (Pi)^((1)/(2)) (2)^(mu - 1)* (z)^(- mu)* GAMMA(mu +(1)/(2))(BesselJ(mu, (1)/(2)z))^(2)	Integrate[BesselJ[\[Mu], zSin[\[Theta]]](Sin[\[Theta]])^\[Mu](Cos[\[Theta]])^(2\[Mu]), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == (Pi)^(Divide[1,2]) (2)^(\[Mu]- 1)* (z)^(- \[Mu])* Gamma[\[Mu]+Divide[1,2]](BesselJ[\[Mu], Divide[1,2]z])^(2)	Successful	Aborted	-	Successful [Tested: 35]
10.22.E21	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{\mu}\left(z\sin\theta% \right)(\sin\theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta=\pi^{\frac{1}{2}}% 2^{\mu-1}z^{-\mu}\\Gamma\left(\mu+\tfrac{1}{2}\right)J_{\mu}\left(\tfrac{1}{2% }z\right)Y_{\mu}\left(\tfrac{1}{2}z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselY{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu}(\cos@@{\theta})^{2\mu}\diff{\theta} = \pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\\EulerGamma@{\mu+\tfrac{1}{2}}\BesselJ{\mu}@{\tfrac{1}{2}z}\BesselY{\mu}@{\tfrac{1}{2}z}`	${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re((\mu)+k+1)>0,\Re(\mu+% \tfrac{1}{2})>0,\Re((-(\mu))+k+1)>0}}$	int(BesselY(mu, zsin(theta))(sin(theta))^(mu)(cos(theta))^(2mu), theta = 0..(1)/(2)Pi) = (Pi)^((1)/(2)) (2)^(mu - 1)* (z)^(- mu)* GAMMA(mu +(1)/(2))BesselJ(mu, (1)/(2)z)BesselY(mu, (1)/(2)z)	Integrate[BesselY[\[Mu], zSin[\[Theta]]](Sin[\[Theta]])^\[Mu](Cos[\[Theta]])^(2\[Mu]), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == (Pi)^(Divide[1,2]) (2)^(\[Mu]- 1)* (z)^(- \[Mu])* Gamma[\[Mu]+Divide[1,2]]BesselJ[\[Mu], Divide[1,2]z]BesselY[\[Mu], Divide[1,2]z]	Successful	Aborted	-	Skipped - Because timed out
10.22.E22	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}% \theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\mu+1}(\cos% \theta)^{2\nu+1}\mathrm{d}\theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)% \Gamma\left(\nu+\tfrac{1}{2}\right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8% \pi z)^{\frac{1}{2}}\Gamma\left(\mu+\nu+1\right)}}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}(\sin@@{\theta})^{2\mu+1}(\cos@@{\theta})^{2\nu+1}\diff{\theta} = \frac{\EulerGamma@{\mu+\tfrac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}\BesselJ{\mu+\nu+\frac{1}{2}}@{z}}{(8\pi z)^{\frac{1}{2}}\EulerGamma@{\mu+\nu+1}}`	${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2},\Re((\mu% )+k+1)>0,\Re(\nu+k+1)>0,\Re((\mu+\nu+\frac{1}{2})+k+1)>0,\Re(\mu+\tfrac{1}{2})% >0,\Re(\nu+\tfrac{1}{2})>0,\Re(\mu+\nu+1)>0}}$	int(BesselJ(mu, z(sin(theta))^(2))BesselJ(nu, z(cos(theta))^(2))(sin(theta))^(2mu + 1)(cos(theta))^(2nu + 1), theta = 0..(1)/(2)Pi) = (GAMMA(mu +(1)/(2))GAMMA(nu +(1)/(2))BesselJ(mu + nu +(1)/(2), z))/((8Piz)^((1)/(2))* GAMMA(mu + nu + 1))	Integrate[BesselJ[\[Mu], z(Sin[\[Theta]])^(2)]BesselJ[\[Nu], z(Cos[\[Theta]])^(2)](Sin[\[Theta]])^(2\[Mu]+ 1)(Cos[\[Theta]])^(2\[Nu]+ 1), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[Gamma[\[Mu]+Divide[1,2]]Gamma[\[Nu]+Divide[1,2]]BesselJ[\[Mu]+ \[Nu]+Divide[1,2], z],(8Piz)^(Divide[1,2])* Gamma[\[Mu]+ \[Nu]+ 1]]	Error	Aborted	-	Skipped - Because timed out
10.22.E23	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}% \theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\alpha-1}\sec% \theta\mathrm{d}\theta=\frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{% \alpha-1}}{\nu\Gamma\left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z% \right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}(\sin@@{\theta})^{2\alpha-1}\sec@@{\theta}\diff{\theta} = \frac{(\mu+\nu+\alpha)\EulerGamma@{\mu+\alpha}2^{\alpha-1}}{\nu\EulerGamma@{\mu+1}z^{\alpha}}\BesselJ{\mu+\nu+\alpha}@{z}`	${\displaystyle{\displaystyle\Re\left(\mu+\alpha\right)>0,\Re\nu>0,\Re((\mu)+k+% 1)>0,\Re(\nu+k+1)>0,\Re((\mu+\nu+\alpha)+k+1)>0,\Re(\mu+\alpha)>0,\Re(\mu+1)>0}}$	int(BesselJ(mu, z(sin(theta))^(2))BesselJ(nu, z(cos(theta))^(2))(sin(theta))^(2alpha - 1) sec(theta), theta = 0..(1)/(2)Pi) = ((mu + nu + alpha)GAMMA(mu + alpha)(2)^(alpha - 1))/(nuGAMMA(mu + 1)(z)^(alpha))BesselJ(mu + nu + alpha, z)	Integrate[BesselJ[\[Mu], z(Sin[\[Theta]])^(2)]BesselJ[\[Nu], z(Cos[\[Theta]])^(2)](Sin[\[Theta]])^(2\[Alpha]- 1) Sec[\[Theta]], {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[(\[Mu]+ \[Nu]+ \[Alpha])Gamma[\[Mu]+ \[Alpha]](2)^(\[Alpha]- 1),\[Nu]Gamma[\[Mu]+ 1](z)^\[Alpha]]BesselJ[\[Mu]+ \[Nu]+ \[Alpha], z]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.22.E24	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}% \theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)\cot\theta\mathrm{d}\theta=% \tfrac{1}{2}\mu^{-1}J_{\mu+\nu}\left(z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}\cot@@{\theta}\diff{\theta} = \tfrac{1}{2}\mu^{-1}\BesselJ{\mu+\nu}@{z}`	${\displaystyle{\displaystyle\Re\mu>0,\Re\nu>-1,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0% ,\Re((\mu+\nu)+k+1)>0}}$	int(BesselJ(mu, z(sin(theta))^(2))BesselJ(nu, z(cos(theta))^(2))cot(theta), theta = 0..(1)/(2)Pi) = (1)/(2)(mu)^(- 1)* BesselJ(mu + nu, z)	Integrate[BesselJ[\[Mu], z(Sin[\[Theta]])^(2)]BesselJ[\[Nu], z(Cos[\[Theta]])^(2)]Cot[\[Theta]], {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[1,2]\[Mu]^(- 1)* BesselJ[\[Mu]+ \[Nu], z]	Failure	Aborted	Skipped - Because timed out	Skip - No test values generated
10.22.E25	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)I_{\nu}\left(z\cos\theta\right)(\tan\theta)^{\mu+1}\mathrm{d}\theta=% \frac{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{1}{2}z)^{\mu}}% {2\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}\left(z\right)}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}\modBesselI{\nu}@{z\cos@@{\theta}}(\tan@@{\theta})^{\mu+1}\diff{\theta} = \frac{\EulerGamma@{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}(\tfrac{1}{2}z)^{\mu}}{2\EulerGamma@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}}\BesselJ{\nu}@{z}`	${\displaystyle{\displaystyle\Re\nu>\Re\mu,\Re\mu>-1,\Re((\mu)+k+1)>0,\Re(\nu+k% +1)>0,\Re(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu)>0,\Re(\tfrac{1}{2}\nu+\tfrac{1}{2}% \mu+1)>0}}$	int(BesselJ(mu, zsin(theta))BesselI(nu, zcos(theta))(tan(theta))^(mu + 1), theta = 0..(1)/(2)Pi) = (GAMMA((1)/(2)nu -(1)/(2)mu)((1)/(2)z)^(mu))/(2GAMMA((1)/(2)nu +(1)/(2)mu + 1))*BesselJ(nu, z)	Integrate[BesselJ[\[Mu], zSin[\[Theta]]]BesselI[\[Nu], zCos[\[Theta]]](Tan[\[Theta]])^(\[Mu]+ 1), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[Gamma[Divide[1,2]\[Nu]-Divide[1,2]\[Mu]](Divide[1,2]z)^\[Mu],2Gamma[Divide[1,2]\[Nu]+Divide[1,2]\[Mu]+ 1]]*BesselJ[\[Nu], z]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.22.E26	${\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)J_{\nu}\left(\zeta\cos\theta\right)(\sin\theta)^{\mu+1}(\cos\theta)^{% \nu+1}\mathrm{d}\theta=\frac{z^{\mu}\zeta^{\nu}J_{\mu+\nu+1}\left(\sqrt{\zeta^% {2}+z^{2}}\right)}{(\zeta^{2}+z^{2})^{\frac{1}{2}(\mu+\nu+1)}}}}$ `\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}\BesselJ{\nu}@{\zeta\cos@@{\theta}}(\sin@@{\theta})^{\mu+1}(\cos@@{\theta})^{\nu+1}\diff{\theta} = \frac{z^{\mu}\zeta^{\nu}\BesselJ{\mu+\nu+1}@{\sqrt{\zeta^{2}+z^{2}}}}{(\zeta^{2}+z^{2})^{\frac{1}{2}(\mu+\nu+1)}}`	${\displaystyle{\displaystyle\Re\mu>-1,\Re\nu>-1,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>% 0,\Re((\mu+\nu+1)+k+1)>0}}$	int(BesselJ(mu, zsin(theta))BesselJ(nu, zetacos(theta))(sin(theta))^(mu + 1)(cos(theta))^(nu + 1), theta = 0..(1)/(2)Pi) = ((z)^(mu)* (zeta)^(nu)* BesselJ(mu + nu + 1, sqrt((zeta)^(2)+ (z)^(2))))/(((zeta)^(2)+ (z)^(2))^((1)/(2)*(mu + nu + 1)))	Integrate[BesselJ[\[Mu], zSin[\[Theta]]]BesselJ[\[Nu], \[Zeta]Cos[\[Theta]]](Sin[\[Theta]])^(\[Mu]+ 1)(Cos[\[Theta]])^(\[Nu]+ 1), {\[Theta], 0, Divide[1,2]Pi}, GenerateConditions->None] == Divide[(z)^\[Mu]* \[Zeta]^\[Nu]* BesselJ[\[Mu]+ \[Nu]+ 1, Sqrt[\[Zeta]^(2)+ (z)^(2)]],(\[Zeta]^(2)+ (z)^(2))^(Divide[1,2]*(\[Mu]+ \[Nu]+ 1))]	Error	Aborted	-	Skipped - Because timed out
10.22.E27	${\displaystyle{\displaystyle\int_{0}^{x}t{J_{\nu-1}^{2}}\left(t\right)\mathrm{% d}t=2\sum_{k=0}^{\infty}(\nu+2k){J_{\nu+2k}^{2}}\left(x\right)}}$ `\int_{0}^{x}t\BesselJ{\nu-1}^{2}@{t}\diff{t} = 2\sum_{k=0}^{\infty}(\nu+2k)\BesselJ{\nu+2k}^{2}@{x}`	${\displaystyle{\displaystyle\Re\nu>0,\Re((\nu-1)+k+1)>0,\Re((\nu+2k)+k+1)>0}}$	int(t(BesselJ(nu - 1, t))^(2), t = 0..x) = 2sum((nu + 2k)(BesselJ(nu + 2*k, x))^(2), k = 0..infinity)	Integrate[t(BesselJ[\[Nu]- 1, t])^(2), {t, 0, x}, GenerateConditions->None] == 2Sum[(\[Nu]+ 2k)(BesselJ[\[Nu]+ 2*k, x])^(2), {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 15]	Successful [Tested: 15]
10.22.E28	${\displaystyle{\displaystyle\int_{0}^{x}t\left({J_{\nu-1}^{2}}\left(t\right)-{% J_{\nu+1}^{2}}\left(t\right)\right)\mathrm{d}t=2\nu{J_{\nu}^{2}}\left(x\right)}}$ `\int_{0}^{x}t\left(\BesselJ{\nu-1}^{2}@{t}-\BesselJ{\nu+1}^{2}@{t}\right)\diff{t} = 2\nu\BesselJ{\nu}^{2}@{x}`	${\displaystyle{\displaystyle\Re\nu>0,\Re((\nu-1)+k+1)>0,\Re((\nu+1)+k+1)>0,\Re% (\nu+k+1)>0}}$	int(t((BesselJ(nu - 1, t))^(2)- (BesselJ(nu + 1, t))^(2)), t = 0..x) = 2nu*(BesselJ(nu, x))^(2)	Integrate[t((BesselJ[\[Nu]- 1, t])^(2)- (BesselJ[\[Nu]+ 1, t])^(2)), {t, 0, x}, GenerateConditions->None] == 2\[Nu]*(BesselJ[\[Nu], x])^(2)	Successful	Successful	-	Successful [Tested: 15]
10.22.E29	${\displaystyle{\displaystyle\int_{0}^{x}t{J_{0}^{2}}\left(t\right)\mathrm{d}t=% \tfrac{1}{2}x^{2}\left({J_{0}^{2}}\left(x\right)+{J_{1}^{2}}\left(x\right)% \right)}}$ `\int_{0}^{x}t\BesselJ{0}^{2}@{t}\diff{t} = \tfrac{1}{2}x^{2}\left(\BesselJ{0}^{2}@{x}+\BesselJ{1}^{2}@{x}\right)`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0}}$	int(t(BesselJ(0, t))^(2), t = 0..x) = (1)/(2)(x)^(2)*((BesselJ(0, x))^(2)+ (BesselJ(1, x))^(2))	Integrate[t(BesselJ[0, t])^(2), {t, 0, x}, GenerateConditions->None] == Divide[1,2](x)^(2)*((BesselJ[0, x])^(2)+ (BesselJ[1, x])^(2))	Successful	Successful	-	Successful [Tested: 3]
10.22.E30	${\displaystyle{\displaystyle\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t% \right)\mathrm{d}t=\tfrac{1}{2}\left(1-{J_{0}^{2}}\left(x\right)\right)-\sum_{% k=1}^{n}{J_{k}^{2}}\left(x\right)}}$ `\int_{0}^{x}\BesselJ{n}@{t}\BesselJ{n+1}@{t}\diff{t} = \tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x}`	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re((n+1)+k+1)>0,\Re(0+k+1)>0,\Re(k+k% +1)>0}}$	int(BesselJ(n, t)BesselJ(n + 1, t), t = 0..x) = (1)/(2)(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n)	Integrate[BesselJ[n, t]BesselJ[n + 1, t], {t, 0, x}, GenerateConditions->None] == Divide[1,2](1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 3]	Failed [2 / 3] Result: Plus[-0.6308420033135872, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[2, ], Power[1.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[1.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[1.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 1.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2], Times[Power[1.5, -2], Power[Plus[Times[-1, 1.5, BesselJ[0, 1.5]], Times[2, BesselJ[1, 1.5]]], 2]]]]}]][4.0]], {Rule[n, 3], Rule[x, 1.5]} Result: Plus[-0.9403627636501156, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[2, ], Power[0.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[0.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[0.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[0.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[0.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 0.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2], Times[Power[0.5, -2], Power[Plus[Times[-1, 0.5, BesselJ[0, 0.5]], Times[2, BesselJ[1, 0.5]]], 2]]]]}]][4.0]], {Rule[n, 3], Rule[x, 0.5]}
10.22.E30	${\displaystyle{\displaystyle\tfrac{1}{2}\left(1-{J_{0}^{2}}\left(x\right)% \right)-\sum_{k=1}^{n}{J_{k}^{2}}\left(x\right)=\sum_{k=n+1}^{\infty}{J_{k}^{2% }}\left(x\right)}}$ `\tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x} = \sum_{k=n+1}^{\infty}\BesselJ{k}^{2}@{x}`	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re((n+1)+k+1)>0,\Re(0+k+1)>0,\Re(k+k% +1)>0}}$	(1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n) = sum((BesselJ(k, x))^(2), k = n + 1..infinity)	Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}, GenerateConditions->None] == Sum[(BesselJ[k, x])^(2), {k, n + 1, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 3]	Failed [3 / 3] Result: Plus[0.6309837827773054, Times[-1.0, NSum[Power[BesselJ[k, 1.5], 2] Test Values: {k, 4, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], Power[1.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[1.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[1.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 1.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2], Times[Power[1.5, -2], Power[Plus[Times[-1, 1.5, BesselJ[0, 1.5]], Times[2, BesselJ[1, 1.5]]], 2]]]]}]][4.0]]], {Ru<syntaxhighlight lang=mathematica>Result: Plus[0.9403627895513045, Times[-1.0, NSum[Power[BesselJ[k, 0.5], 2] Test Values: {k, 4, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], Power[0.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[0.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[0.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[0.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[0.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 0.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2], Times[Power[0.5, -2], Power[Plus[Times[-1, 0.5, BesselJ[0, 0.5]], Times[2, BesselJ[1, 0.5]]], 2]]]]}]][4.0]]], {Rule[n, 3], Rule[x, 0.5]} ... skip entries to safe data
10.22.E31	${\displaystyle{\displaystyle\int_{0}^{x}J_{\mu}\left(t\right)J_{\nu}\left(x-t% \right)\mathrm{d}t=2\sum_{k=0}^{\infty}(-1)^{k}J_{\mu+\nu+2k+1}\left(x\right)}}$ `\int_{0}^{x}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = 2\sum_{k=0}^{\infty}(-1)^{k}\BesselJ{\mu+\nu+2k+1}@{x}`	${\displaystyle{\displaystyle\Re\mu>-1,\Re\nu>-1,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>% 0,\Re((\mu+\nu+2k+1)+k+1)>0}}$	int(BesselJ(mu, t)BesselJ(nu, x - t), t = 0..x) = 2sum((- 1)^(k)* BesselJ(mu + nu + 2*k + 1, x), k = 0..infinity)	Integrate[BesselJ[\[Mu], t]BesselJ[\[Nu], x - t], {t, 0, x}, GenerateConditions->None] == 2Sum[(- 1)^(k)* BesselJ[\[Mu]+ \[Nu]+ 2*k + 1, x], {k, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Skip - No test values generated
10.22.E32	${\displaystyle{\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)J_{1-\nu}\left(x-% t\right)\mathrm{d}t=J_{0}\left(x\right)-\cos x}}$ `\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{1-\nu}@{x-t}\diff{t} = \BesselJ{0}@{x}-\cos@@{x}`	${\displaystyle{\displaystyle-1<\Re\nu,\Re\nu<2,\Re(\nu+k+1)>0,\Re((1-\nu)+k+1)% >0,\Re(0+k+1)>0}}$	int(BesselJ(nu, t)*BesselJ(1 - nu, x - t), t = 0..x) = BesselJ(0, x)- cos(x)	Integrate[BesselJ[\[Nu], t]*BesselJ[1 - \[Nu], x - t], {t, 0, x}, GenerateConditions->None] == BesselJ[0, x]- Cos[x]	Failure	Failure	Manual Skip!	Skipped - Because timed out
10.22.E33	${\displaystyle{\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)J_{-\nu}\left(x-t% \right)\mathrm{d}t=\sin x}}$ `\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{-\nu}@{x-t}\diff{t} = \sin@@{x}`	${\displaystyle{\displaystyle\|\Re\nu\|<1,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	int(BesselJ(nu, t)*BesselJ(- nu, x - t), t = 0..x) = sin(x)	Integrate[BesselJ[\[Nu], t]*BesselJ[- \[Nu], x - t], {t, 0, x}, GenerateConditions->None] == Sin[x]	Failure	Failure	Manual Skip!	Skipped - Because timed out
10.22.E34	${\displaystyle{\displaystyle\int_{0}^{x}t^{-1}J_{\mu}\left(t\right)J_{\nu}% \left(x-t\right)\mathrm{d}t=\frac{J_{\mu+\nu}\left(x\right)}{\mu}}}$ `\int_{0}^{x}t^{-1}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = \frac{\BesselJ{\mu+\nu}@{x}}{\mu}`	${\displaystyle{\displaystyle\Re\mu>0,\Re\nu>-1,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0% ,\Re((\mu+\nu)+k+1)>0}}$	int((t)^(- 1)* BesselJ(mu, t)*BesselJ(nu, x - t), t = 0..x) = (BesselJ(mu + nu, x))/(mu)	Integrate[(t)^(- 1)* BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t], {t, 0, x}, GenerateConditions->None] == Divide[BesselJ[\[Mu]+ \[Nu], x],\[Mu]]	Failure	Failure	Manual Skip!	Skip - No test values generated
10.22.E35	${\displaystyle{\displaystyle\int_{0}^{x}\frac{J_{\mu}\left(t\right)J_{\nu}% \left(x-t\right)\mathrm{d}t}{t(x-t)}=\frac{(\mu+\nu)J_{\mu+\nu}\left(x\right)}% {\mu\nu x}}}$ `\int_{0}^{x}\frac{\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t}}{t(x-t)} = \frac{(\mu+\nu)\BesselJ{\mu+\nu}@{x}}{\mu\nu x}`	${\displaystyle{\displaystyle\Re\mu>0,\Re\nu>0,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,% \Re((\mu+\nu)+k+1)>0}}$	int((BesselJ(mu, t)BesselJ(nu, x - t))/(t(x - t)), t = 0..x) = ((mu + nu)BesselJ(mu + nu, x))/(munu*x)	Integrate[Divide[BesselJ[\[Mu], t]BesselJ[\[Nu], x - t],t(x - t)], {t, 0, x}, GenerateConditions->None] == Divide[(\[Mu]+ \[Nu])BesselJ[\[Mu]+ \[Nu], x],\[Mu]\[Nu]*x]	Error	Failure	-	Skip - No test values generated
10.22.E36	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{x}(x-% t)^{\alpha-1}J_{\nu}\left(t\right)\mathrm{d}t=2^{\alpha}\sum_{k=0}^{\infty}% \frac{(\alpha)_{k}}{k!}J_{\nu+\alpha+2k}\left(x\right)}}$ `\frac{1}{\EulerGamma@{\alpha}}\int_{0}^{x}(x-t)^{\alpha-1}\BesselJ{\nu}@{t}\diff{t} = 2^{\alpha}\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{k!}\BesselJ{\nu+\alpha+2k}@{x}`	${\displaystyle{\displaystyle\Re\alpha>0,\Re\nu\geq 0,\Re(\nu+k+1)>0,\Re((\nu+% \alpha+2k)+k+1)>0,\Re(\alpha)>0}}$	(1)/(GAMMA(alpha))int((x - t)^(alpha - 1) BesselJ(nu, t), t = 0..x) = (2)^(alpha)* sum((alpha[k])/(factorial(k))BesselJ(nu + alpha + 2k, x), k = 0..infinity)	Divide[1,Gamma[\[Alpha]]]Integrate[(x - t)^(\[Alpha]- 1) BesselJ[\[Nu], t], {t, 0, x}, GenerateConditions->None] == (2)^\[Alpha]* Sum[Divide[Subscript[\[Alpha], k],(k)!]BesselJ[\[Nu]+ \[Alpha]+ 2k, x], {k, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Skip - No test values generated
10.22.E37	${\displaystyle{\displaystyle\int_{0}^{1}tJ_{\nu}\left(j_{\nu,\ell}t\right)J_{% \nu}\left(j_{\nu,m}t\right)\mathrm{d}t=\tfrac{1}{2}\left(J_{\nu}'\left(j_{\nu,% \ell}\right)\right)^{2}\delta_{\ell,m}}}$ `\int_{0}^{1}t\BesselJ{\nu}@{j_{\nu,\ell}t}\BesselJ{\nu}@{j_{\nu,m}t}\diff{t} = \tfrac{1}{2}\left(\BesselJ{\nu}'@{j_{\nu,\ell}}\right)^{2}\Kroneckerdelta{\ell}{m}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	int(tBesselJ(nu, j[nu , ell]t)BesselJ(nu, j[nu , m]t), t = 0..1) = (1)/(2)(diff( BesselJ(nu, j[nu , ell]), j[nu , ell]$(1) ))^(2) KroneckerDelta[ell, m]	Integrate[tBesselJ[\[Nu], Subscript[j, \[Nu], \[ScriptL]]t]BesselJ[\[Nu], Subscript[j, \[Nu], m]t], {t, 0, 1}, GenerateConditions->None] == Divide[1,2](D[BesselJ[\[Nu], Subscript[j, \[Nu], \[ScriptL]]], {Subscript[j, \[Nu], \[ScriptL]], 1}])^(2) KroneckerDelta[\[ScriptL], m]	Failure	Failure	Error	Failed [300 / 300] Result: Indeterminate Test Values: {Rule[m, 1], Rule[ℓ, 1], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, ν, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, ν, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[m, 1], Rule[ℓ, 2], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, ν, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, ν, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data

@@ Line 309: / Line 309: @@
 | [https://dlmf.nist.gov/10.14.E2 10.14.E2] || [[Item:Q3139|<math>0 < \BesselJ{\nu}@{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>0 < \BesselJ{\nu}@{\nu}</syntaxhighlight> || <math>\nu > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>0 < BesselJ(nu, nu)</syntaxhighlight> || <syntaxhighlight lang=mathematica>0 < BesselJ[\[Nu], \[Nu]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/10.14.E2 10.14.E2] || [[Item:Q3139|<math>\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}</syntaxhighlight> || <math>\nu > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\tfrac{2}{3})} > 0</math> || <syntaxhighlight lang=mathematica>BesselJ(nu, nu) < ((2)^((1)/(3)))/((3)^((2)/(3))* GAMMA((2)/(3))*(nu)^((1)/(3)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselJ[\[Nu], \[Nu]] < Divide[(2)^(Divide[1,3]),(3)^(Divide[2,3])* Gamma[Divide[2,3]]*\[Nu]^(Divide[1,3])]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/10.14.E2 10.14.E2] || [[Item:Q3139|<math>\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}</syntaxhighlight> || <math>\nu > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselJ(nu, nu) < ((2)^((1)/(3)))/((3)^((2)/(3))* GAMMA((2)/(3))*(nu)^((1)/(3)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselJ[\[Nu], \[Nu]] < Divide[(2)^(Divide[1,3]),(3)^(Divide[2,3])* Gamma[Divide[2,3]]*\[Nu]^(Divide[1,3])]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
 | [https://dlmf.nist.gov/10.14.E3 10.14.E3] || [[Item:Q3140|<math>|\BesselJ{n}@{z}| \leq e^{|\imagpart@@{z}|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\BesselJ{n}@{z}| \leq e^{|\imagpart@@{z}|}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>abs(BesselJ(n, z)) <= exp(abs(Im(z)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[BesselJ[n, z]] <= Exp[Abs[Im[z]]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
@@ Line 499: / Line 499: @@
 |- style="background: #dfe6e9;"
 | [https://dlmf.nist.gov/10.21#Ex58 10.21#Ex58] || [[Item:Q3370|<math>r = \frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^{5}-(\mu-1)(\mu^{2}-114\mu+1073)}{5(4\lambda)^{5}(\lambda-1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r = \frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^{5}-(\mu-1)(\mu^{2}-114\mu+1073)}{5(4\lambda)^{5}(\lambda-1)}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(((mu - 1)*((mu)^(2)- 114*mu + 1073)*((lambda)^(5)- 1))/(5*(4*lambda)^(5)*(lambda - 1))) = (((mu)^(3)+ 185*(mu)^(2)- 2053*mu + 1899)*(lambda)^(5)-(mu - 1)*((mu)^(2)- 114*mu + 1073))/(5*(4*lambda)^(5)*(lambda - 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[(\[Mu]- 1)*(\[Mu]^(2)- 114*\[Mu]+ 1073)*(\[Lambda]^(5)- 1),5*(4*\[Lambda])^(5)*(\[Lambda]- 1)]) == Divide[(\[Mu]^(3)+ 185*\[Mu]^(2)- 2053*\[Mu]+ 1899)*\[Lambda]^(5)-(\[Mu]- 1)*(\[Mu]^(2)- 114*\[Mu]+ 1073),5*(4*\[Lambda])^(5)*(\[Lambda]- 1)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
 |-
 | [https://dlmf.nist.gov/10.22.E8 10.22.E8] || [[Item:Q3382|<math>\int_{0}^{x}\BesselJ{\nu}@{t}\diff{t} = 2\sum_{k=0}^{\infty}\BesselJ{\nu+2k+1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{\nu}@{t}\diff{t} = 2\sum_{k=0}^{\infty}\BesselJ{\nu+2k+1}@{x}</syntaxhighlight> || <math>\realpart@@{\nu} > -1, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+2k+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(nu, t), t = 0..x) = 2*sum(BesselJ(nu + 2*k + 1, x), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Nu], t], {t, 0, x}, GenerateConditions->None] == 2*Sum[BesselJ[\[Nu]+ 2*k + 1, x], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 24]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.277492396
 Test Values: {nu = -1/2, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1653166018
 Test Values: {nu = 1/2, x = 3/2}</syntaxhighlight><br></div></div> || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E9 10.22.E9] || [[Item:Q3383|<math>\int_{0}^{x}\BesselJ{2n}@{t}\diff{t} = \int_{0}^{x}\BesselJ{0}@{t}\diff{t}-2\sum_{k=0}^{n-1}\BesselJ{2k+1}@{x},\quad\int_{0}^{x}\BesselJ{2n+1}@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{2n}@{t}\diff{t} = \int_{0}^{x}\BesselJ{0}@{t}\diff{t}-2\sum_{k=0}^{n-1}\BesselJ{2k+1}@{x},\quad\int_{0}^{x}\BesselJ{2n+1}@{t}\diff{t}</syntaxhighlight> || <math>\realpart@@{((2n)+k+1)} > 0, \realpart@@{(0+k+1)} > 0, \realpart@@{((2k+1)+k+1)} > 0, \realpart@@{((2n+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(2*n, t), t = 0..x) = int(BesselJ(0, t), t = 0..x)- 2*sum(BesselJ(2*k + 1, x), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[2*n, t], {t, 0, x}, GenerateConditions->None] == Integrate[BesselJ[0, t], {t, 0, x}, GenerateConditions->None]- 2*Sum[BesselJ[2*k + 1, x], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Error
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 | [https://dlmf.nist.gov/10.22.E9 10.22.E9] || [[Item:Q3383|<math>\int_{0}^{x}\BesselJ{0}@{t}\diff{t}-2\sum_{k=0}^{n-1}\BesselJ{2k+1}@{x},\quad\int_{0}^{x}\BesselJ{2n+1}@{t}\diff{t} = 1-\BesselJ{0}@{x}-2\sum_{k=1}^{n}\BesselJ{2k}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{0}@{t}\diff{t}-2\sum_{k=0}^{n-1}\BesselJ{2k+1}@{x},\quad\int_{0}^{x}\BesselJ{2n+1}@{t}\diff{t} = 1-\BesselJ{0}@{x}-2\sum_{k=1}^{n}\BesselJ{2k}@{x}</syntaxhighlight> || <math>\realpart@@{((2n)+k+1)} > 0, \realpart@@{(0+k+1)} > 0, \realpart@@{((2k+1)+k+1)} > 0, \realpart@@{((2n+1)+k+1)} > 0, \realpart@@{((2k)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(0, t), t = 0..x)- 2*sum(BesselJ(2*k + 1, x), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[0, t], {t, 0, x}, GenerateConditions->None]- 2*Sum[BesselJ[2*k + 1, x], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Error
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 | [https://dlmf.nist.gov/10.22.E10 10.22.E10] || [[Item:Q3384|<math>\int_{0}^{x}t^{\mu}\BesselJ{\nu}@{t}\diff{t} = x^{\mu}\frac{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k}}{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k}}\BesselJ{\nu+2k+1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t^{\mu}\BesselJ{\nu}@{t}\diff{t} = x^{\mu}\frac{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k}}{\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k}}\BesselJ{\nu+2k+1}@{x}</syntaxhighlight> || <math>\realpart@{\mu+\nu+1} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+2k+1)+k+1)} > 0, \realpart@@{(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})} > 0, \realpart@@{(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2})} > 0, \realpart@@{(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k)} > 0, \realpart@@{(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(mu)* BesselJ(nu, t), t = 0..x) = (x)^(mu)*(GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))* sum(((nu + 2*k + 1)*GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)+ k))/(GAMMA((1)/(2)*nu +(1)/(2)*mu +(3)/(2)+ k))*BesselJ(nu + 2*k + 1, x), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^\[Mu]* BesselJ[\[Nu], t], {t, 0, x}, GenerateConditions->None] == (x)^\[Mu]*Divide[Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]* Sum[Divide[(\[Nu]+ 2*k + 1)*Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]+ k],Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[3,2]+ k]]*BesselJ[\[Nu]+ 2*k + 1, x], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E11 10.22.E11] || [[Item:Q3385|<math>\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = \frac{1}{2}\sum_{k=1}^{\infty}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\BesselJ{k}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = \frac{1}{2}\sum_{k=1}^{\infty}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\BesselJ{k}@{x}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(k+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int((1 - BesselJ(0, t))/(t), t = 0..x) = (1)/(2)*sum((Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselJ(k, x), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}, GenerateConditions->None] == Divide[1,2]*Sum[Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselJ[k, x], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.2622772441151432, Times[-0.5, NSum[Times[Power[0.75, k], BesselJ[k, 1.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
 Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.03100698635091531, Times[-0.5, NSum[Times[Power[0.25, k], BesselJ[k, 0.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
 Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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 | [https://dlmf.nist.gov/10.22.E12 10.22.E12] || [[Item:Q3386|<math>x\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = 2\sum_{k=0}^{\infty}(2k+3)(\digamma@{k+2}-\digamma@{1})\BesselJ{2k+3}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x\int_{0}^{x}\frac{1-\BesselJ{0}@{t}}{t}\diff{t} = 2\sum_{k=0}^{\infty}(2k+3)(\digamma@{k+2}-\digamma@{1})\BesselJ{2k+3}@{x}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{((2k+3)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>x*int((1 - BesselJ(0, t))/(t), t = 0..x) = 2*sum((2*k + 3)*(Psi(k + 2)- Psi(1))*BesselJ(2*k + 3, x), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x*Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}, GenerateConditions->None] == 2*Sum[(2*k + 3)*(PolyGamma[k + 2]- PolyGamma[1])*BesselJ[2*k + 3, x], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 3] || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E12 10.22.E12] || [[Item:Q3386|<math>2\sum_{k=0}^{\infty}(2k+3)(\digamma@{k+2}-\digamma@{1})\BesselJ{2k+3}@{x} = x-2\BesselJ{1}@{x}+2\sum_{k=0}^{\infty}(2k+5)\*(\digamma@{k+3}-\digamma@{1}-1)\BesselJ{2k+5}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\sum_{k=0}^{\infty}(2k+3)(\digamma@{k+2}-\digamma@{1})\BesselJ{2k+3}@{x} = x-2\BesselJ{1}@{x}+2\sum_{k=0}^{\infty}(2k+5)\*(\digamma@{k+3}-\digamma@{1}-1)\BesselJ{2k+5}@{x}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{((2k+3)+k+1)} > 0, \realpart@@{(1+k+1)} > 0, \realpart@@{((2k+5)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>2*sum((2*k + 3)*(Psi(k + 2)- Psi(1))*BesselJ(2*k + 3, x), k = 0..infinity) = x - 2*BesselJ(1, x)+ 2*sum((2*k + 5)*(Psi(k + 3)- Psi(1)- 1)*BesselJ(2*k + 5, x), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*Sum[(2*k + 3)*(PolyGamma[k + 2]- PolyGamma[1])*BesselJ[2*k + 3, x], {k, 0, Infinity}, GenerateConditions->None] == x - 2*BesselJ[1, x]+ 2*Sum[(2*k + 5)*(PolyGamma[k + 3]- PolyGamma[1]- 1)*BesselJ[2*k + 5, x], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Successful [Tested: 3] || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E13 10.22.E13] || [[Item:Q3387|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</syntaxhighlight> || <math>\realpart@@{\nu} > -\tfrac{1}{2}, \realpart@@{((2\nu)+k+1)} > 0, \realpart@@{((\nu+\mu)+k+1)} > 0, \realpart@@{((\nu-\mu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(2*nu, 2*z*cos(theta))*cos(2*mu*theta), theta = 0..(1)/(2)*Pi) = (1)/(2)*Pi*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[2*\[Nu], 2*z*Cos[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[1,2]*Pi*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E14 10.22.E14] || [[Item:Q3388|<math>\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \pi\cos@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \pi\cos@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</syntaxhighlight> || <math>\realpart@@{\nu} > -\tfrac{1}{2}, \realpart@@{((2\nu)+k+1)} > 0, \realpart@@{((\nu+\mu)+k+1)} > 0, \realpart@@{((\nu-\mu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(2*nu, 2*z*sin(theta))*cos(2*mu*theta), theta = 0..Pi) = Pi*cos(mu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[2*\[Nu], 2*z*Sin[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == Pi*Cos[\[Mu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E15 10.22.E15] || [[Item:Q3389|<math>\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\sin@{2\mu\theta}\diff{\theta} = \pi\sin@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\BesselJ{2\nu}@{2z\sin@@{\theta}}\sin@{2\mu\theta}\diff{\theta} = \pi\sin@{\mu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}</syntaxhighlight> || <math>\realpart@@{\nu} > -1, \realpart@@{((2\nu)+k+1)} > 0, \realpart@@{((\nu+\mu)+k+1)} > 0, \realpart@@{((\nu-\mu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(2*nu, 2*z*sin(theta))*sin(2*mu*theta), theta = 0..Pi) = Pi*sin(mu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[2*\[Nu], 2*z*Sin[\[Theta]]]*Sin[2*\[Mu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == Pi*Sin[\[Mu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E16 10.22.E16] || [[Item:Q3390|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}^{2}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}^{2}@{z}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(0, 2*z*sin(theta))*cos(2*n*theta), theta = 0..(1)/(2)*Pi) = (1)/(2)*Pi*(BesselJ(n, z))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[0, 2*z*Sin[\[Theta]]]*Cos[2*n*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[1,2]*Pi*(BesselJ[n, z])^(2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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 | [https://dlmf.nist.gov/10.22.E17 10.22.E17] || [[Item:Q3391|<math>\int_{0}^{\frac{1}{2}\pi}\BesselY{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\cot@{2\nu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}-\tfrac{1}{2}\pi\csc@{2\nu\pi}\BesselJ{\mu-\nu}@{z}\BesselJ{-\mu-\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselY{2\nu}@{2z\cos@@{\theta}}\cos@{2\mu\theta}\diff{\theta} = \tfrac{1}{2}\pi\cot@{2\nu\pi}\BesselJ{\nu+\mu}@{z}\BesselJ{\nu-\mu}@{z}-\tfrac{1}{2}\pi\csc@{2\nu\pi}\BesselJ{\mu-\nu}@{z}\BesselJ{-\mu-\nu}@{z}</syntaxhighlight> || <math>-\tfrac{1}{2} < \realpart@@{\nu}, \realpart@@{\nu} < \tfrac{1}{2}, \realpart@@{((\nu+\mu)+k+1)} > 0, \realpart@@{((\nu-\mu)+k+1)} > 0, \realpart@@{((\mu-\nu)+k+1)} > 0, \realpart@@{((-\mu-\nu)+k+1)} > 0, \realpart@@{((2\nu)+k+1)} > 0, \realpart@@{((-(2\nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselY(2*nu, 2*z*cos(theta))*cos(2*mu*theta), theta = 0..(1)/(2)*Pi) = (1)/(2)*Pi*cot(2*nu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)-(1)/(2)*Pi*csc(2*nu*Pi)*BesselJ(mu - nu, z)*BesselJ(- mu - nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselY[2*\[Nu], 2*z*Cos[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[1,2]*Pi*Cot[2*\[Nu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]-Divide[1,2]*Pi*Csc[2*\[Nu]*Pi]*BesselJ[\[Mu]- \[Nu], z]*BesselJ[- \[Mu]- \[Nu], z]</syntaxhighlight> || Failure || Failure || Error || Skip - No test values generated
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 | [https://dlmf.nist.gov/10.22.E18 10.22.E18] || [[Item:Q3392|<math>\int_{0}^{\frac{1}{2}\pi}\BesselY{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}@{z}\BesselY{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselY{0}@{2z\sin@@{\theta}}\cos@{2n\theta}\diff{\theta} = \tfrac{1}{2}\pi\BesselJ{n}@{z}\BesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0, \realpart@@{(0+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0, \realpart@@{((-n)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselY(0, 2*z*sin(theta))*cos(2*n*theta), theta = 0..(1)/(2)*Pi) = (1)/(2)*Pi*BesselJ(n, z)*BesselY(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselY[0, 2*z*Sin[\[Theta]]]*Cos[2*n*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[1,2]*Pi*BesselJ[n, z]*BesselY[n, z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E19 10.22.E19] || [[Item:Q3393|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu+1}(\cos@@{\theta})^{2\nu+1}\diff{\theta} = 2^{\nu}\EulerGamma@{\nu+1}z^{-\nu-1}\BesselJ{\mu+\nu+1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu+1}(\cos@@{\theta})^{2\nu+1}\diff{\theta} = 2^{\nu}\EulerGamma@{\nu+1}z^{-\nu-1}\BesselJ{\mu+\nu+1}@{z}</syntaxhighlight> || <math>\realpart@@{\mu} > -1, \realpart@@{\nu} > -1, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{((\mu+\nu+1)+k+1)} > 0, \realpart@@{(\nu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*sin(theta))*(sin(theta))^(mu + 1)*(cos(theta))^(2*nu + 1), theta = 0..(1)/(2)*Pi) = (2)^(nu)* GAMMA(nu + 1)*(z)^(- nu - 1)* BesselJ(mu + nu + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*(Sin[\[Theta]])^(\[Mu]+ 1)*(Cos[\[Theta]])^(2*\[Nu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == (2)^\[Nu]* Gamma[\[Nu]+ 1]*(z)^(- \[Nu]- 1)* BesselJ[\[Mu]+ \[Nu]+ 1, z]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 300]
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 | [https://dlmf.nist.gov/10.22.E20 10.22.E20] || [[Item:Q3394|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu}(\cos@@{\theta})^{2\mu}\diff{\theta} = \pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\EulerGamma@{\mu+\tfrac{1}{2}}\BesselJ{\mu}^{2}@{\tfrac{1}{2}z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu}(\cos@@{\theta})^{2\mu}\diff{\theta} = \pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\EulerGamma@{\mu+\tfrac{1}{2}}\BesselJ{\mu}^{2}@{\tfrac{1}{2}z}</syntaxhighlight> || <math>\realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\mu+\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*sin(theta))*(sin(theta))^(mu)*(cos(theta))^(2*mu), theta = 0..(1)/(2)*Pi) = (Pi)^((1)/(2))* (2)^(mu - 1)* (z)^(- mu)* GAMMA(mu +(1)/(2))*(BesselJ(mu, (1)/(2)*z))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*(Sin[\[Theta]])^\[Mu]*(Cos[\[Theta]])^(2*\[Mu]), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == (Pi)^(Divide[1,2])* (2)^(\[Mu]- 1)* (z)^(- \[Mu])* Gamma[\[Mu]+Divide[1,2]]*(BesselJ[\[Mu], Divide[1,2]*z])^(2)</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 35]
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 | [https://dlmf.nist.gov/10.22.E21 10.22.E21] || [[Item:Q3395|<math>\int_{0}^{\frac{1}{2}\pi}\BesselY{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu}(\cos@@{\theta})^{2\mu}\diff{\theta} = \pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\EulerGamma@{\mu+\tfrac{1}{2}}\BesselJ{\mu}@{\tfrac{1}{2}z}\BesselY{\mu}@{\tfrac{1}{2}z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselY{\mu}@{z\sin@@{\theta}}(\sin@@{\theta})^{\mu}(\cos@@{\theta})^{2\mu}\diff{\theta} = \pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\EulerGamma@{\mu+\tfrac{1}{2}}\BesselJ{\mu}@{\tfrac{1}{2}z}\BesselY{\mu}@{\tfrac{1}{2}z}</syntaxhighlight> || <math>\realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\mu+\tfrac{1}{2})} > 0, \realpart@@{((-(\mu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselY(mu, z*sin(theta))*(sin(theta))^(mu)*(cos(theta))^(2*mu), theta = 0..(1)/(2)*Pi) = (Pi)^((1)/(2))* (2)^(mu - 1)* (z)^(- mu)* GAMMA(mu +(1)/(2))*BesselJ(mu, (1)/(2)*z)*BesselY(mu, (1)/(2)*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselY[\[Mu], z*Sin[\[Theta]]]*(Sin[\[Theta]])^\[Mu]*(Cos[\[Theta]])^(2*\[Mu]), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == (Pi)^(Divide[1,2])* (2)^(\[Mu]- 1)* (z)^(- \[Mu])* Gamma[\[Mu]+Divide[1,2]]*BesselJ[\[Mu], Divide[1,2]*z]*BesselY[\[Mu], Divide[1,2]*z]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E22 10.22.E22] || [[Item:Q3396|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}(\sin@@{\theta})^{2\mu+1}(\cos@@{\theta})^{2\nu+1}\diff{\theta} = \frac{\EulerGamma@{\mu+\tfrac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}\BesselJ{\mu+\nu+\frac{1}{2}}@{z}}{(8\pi z)^{\frac{1}{2}}\EulerGamma@{\mu+\nu+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}(\sin@@{\theta})^{2\mu+1}(\cos@@{\theta})^{2\nu+1}\diff{\theta} = \frac{\EulerGamma@{\mu+\tfrac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}\BesselJ{\mu+\nu+\frac{1}{2}}@{z}}{(8\pi z)^{\frac{1}{2}}\EulerGamma@{\mu+\nu+1}}</syntaxhighlight> || <math>\realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{\nu} > -\tfrac{1}{2}, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu+\frac{1}{2})+k+1)} > 0, \realpart@@{(\mu+\tfrac{1}{2})} > 0, \realpart@@{(\nu+\tfrac{1}{2})} > 0, \realpart@@{(\mu+\nu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*(sin(theta))^(2))*BesselJ(nu, z*(cos(theta))^(2))*(sin(theta))^(2*mu + 1)*(cos(theta))^(2*nu + 1), theta = 0..(1)/(2)*Pi) = (GAMMA(mu +(1)/(2))*GAMMA(nu +(1)/(2))*BesselJ(mu + nu +(1)/(2), z))/((8*Pi*z)^((1)/(2))* GAMMA(mu + nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*(Sin[\[Theta]])^(2)]*BesselJ[\[Nu], z*(Cos[\[Theta]])^(2)]*(Sin[\[Theta]])^(2*\[Mu]+ 1)*(Cos[\[Theta]])^(2*\[Nu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]+Divide[1,2]]*BesselJ[\[Mu]+ \[Nu]+Divide[1,2], z],(8*Pi*z)^(Divide[1,2])* Gamma[\[Mu]+ \[Nu]+ 1]]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E23 10.22.E23] || [[Item:Q3397|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}(\sin@@{\theta})^{2\alpha-1}\sec@@{\theta}\diff{\theta} = \frac{(\mu+\nu+\alpha)\EulerGamma@{\mu+\alpha}2^{\alpha-1}}{\nu\EulerGamma@{\mu+1}z^{\alpha}}\BesselJ{\mu+\nu+\alpha}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}(\sin@@{\theta})^{2\alpha-1}\sec@@{\theta}\diff{\theta} = \frac{(\mu+\nu+\alpha)\EulerGamma@{\mu+\alpha}2^{\alpha-1}}{\nu\EulerGamma@{\mu+1}z^{\alpha}}\BesselJ{\mu+\nu+\alpha}@{z}</syntaxhighlight> || <math>\realpart@{\mu+\alpha} > 0, \realpart@@{\nu} > 0, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu+\alpha)+k+1)} > 0, \realpart@@{(\mu+\alpha)} > 0, \realpart@@{(\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*(sin(theta))^(2))*BesselJ(nu, z*(cos(theta))^(2))*(sin(theta))^(2*alpha - 1)* sec(theta), theta = 0..(1)/(2)*Pi) = ((mu + nu + alpha)*GAMMA(mu + alpha)*(2)^(alpha - 1))/(nu*GAMMA(mu + 1)*(z)^(alpha))*BesselJ(mu + nu + alpha, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*(Sin[\[Theta]])^(2)]*BesselJ[\[Nu], z*(Cos[\[Theta]])^(2)]*(Sin[\[Theta]])^(2*\[Alpha]- 1)* Sec[\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[(\[Mu]+ \[Nu]+ \[Alpha])*Gamma[\[Mu]+ \[Alpha]]*(2)^(\[Alpha]- 1),\[Nu]*Gamma[\[Mu]+ 1]*(z)^\[Alpha]]*BesselJ[\[Mu]+ \[Nu]+ \[Alpha], z]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E24 10.22.E24] || [[Item:Q3398|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}\cot@@{\theta}\diff{\theta} = \tfrac{1}{2}\mu^{-1}\BesselJ{\mu+\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin^{2}@@{\theta}}\BesselJ{\nu}@{z\cos^{2}@@{\theta}}\cot@@{\theta}\diff{\theta} = \tfrac{1}{2}\mu^{-1}\BesselJ{\mu+\nu}@{z}</syntaxhighlight> || <math>\realpart@@{\mu} > 0, \realpart@@{\nu} > -1, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*(sin(theta))^(2))*BesselJ(nu, z*(cos(theta))^(2))*cot(theta), theta = 0..(1)/(2)*Pi) = (1)/(2)*(mu)^(- 1)* BesselJ(mu + nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*(Sin[\[Theta]])^(2)]*BesselJ[\[Nu], z*(Cos[\[Theta]])^(2)]*Cot[\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[1,2]*\[Mu]^(- 1)* BesselJ[\[Mu]+ \[Nu], z]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skip - No test values generated
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 | [https://dlmf.nist.gov/10.22.E25 10.22.E25] || [[Item:Q3399|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}\modBesselI{\nu}@{z\cos@@{\theta}}(\tan@@{\theta})^{\mu+1}\diff{\theta} = \frac{\EulerGamma@{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}(\tfrac{1}{2}z)^{\mu}}{2\EulerGamma@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}}\BesselJ{\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}\modBesselI{\nu}@{z\cos@@{\theta}}(\tan@@{\theta})^{\mu+1}\diff{\theta} = \frac{\EulerGamma@{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}(\tfrac{1}{2}z)^{\mu}}{2\EulerGamma@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}}\BesselJ{\nu}@{z}</syntaxhighlight> || <math>\realpart@@{\nu} > \realpart@@{\mu}, \realpart@@{\mu} > -1, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu)} > 0, \realpart@@{(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*sin(theta))*BesselI(nu, z*cos(theta))*(tan(theta))^(mu + 1), theta = 0..(1)/(2)*Pi) = (GAMMA((1)/(2)*nu -(1)/(2)*mu)*((1)/(2)*z)^(mu))/(2*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))*BesselJ(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*BesselI[\[Nu], z*Cos[\[Theta]]]*(Tan[\[Theta]])^(\[Mu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]]*(Divide[1,2]*z)^\[Mu],2*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]]*BesselJ[\[Nu], z]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E26 10.22.E26] || [[Item:Q3400|<math>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}\BesselJ{\nu}@{\zeta\cos@@{\theta}}(\sin@@{\theta})^{\mu+1}(\cos@@{\theta})^{\nu+1}\diff{\theta} = \frac{z^{\mu}\zeta^{\nu}\BesselJ{\mu+\nu+1}@{\sqrt{\zeta^{2}+z^{2}}}}{(\zeta^{2}+z^{2})^{\frac{1}{2}(\mu+\nu+1)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\frac{1}{2}\pi}\BesselJ{\mu}@{z\sin@@{\theta}}\BesselJ{\nu}@{\zeta\cos@@{\theta}}(\sin@@{\theta})^{\mu+1}(\cos@@{\theta})^{\nu+1}\diff{\theta} = \frac{z^{\mu}\zeta^{\nu}\BesselJ{\mu+\nu+1}@{\sqrt{\zeta^{2}+z^{2}}}}{(\zeta^{2}+z^{2})^{\frac{1}{2}(\mu+\nu+1)}}</syntaxhighlight> || <math>\realpart@@{\mu} > -1, \realpart@@{\nu} > -1, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, z*sin(theta))*BesselJ(nu, zeta*cos(theta))*(sin(theta))^(mu + 1)*(cos(theta))^(nu + 1), theta = 0..(1)/(2)*Pi) = ((z)^(mu)* (zeta)^(nu)* BesselJ(mu + nu + 1, sqrt((zeta)^(2)+ (z)^(2))))/(((zeta)^(2)+ (z)^(2))^((1)/(2)*(mu + nu + 1)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*BesselJ[\[Nu], \[Zeta]*Cos[\[Theta]]]*(Sin[\[Theta]])^(\[Mu]+ 1)*(Cos[\[Theta]])^(\[Nu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None] == Divide[(z)^\[Mu]* \[Zeta]^\[Nu]* BesselJ[\[Mu]+ \[Nu]+ 1, Sqrt[\[Zeta]^(2)+ (z)^(2)]],(\[Zeta]^(2)+ (z)^(2))^(Divide[1,2]*(\[Mu]+ \[Nu]+ 1))]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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 | [https://dlmf.nist.gov/10.22.E27 10.22.E27] || [[Item:Q3401|<math>\int_{0}^{x}t\BesselJ{\nu-1}^{2}@{t}\diff{t} = 2\sum_{k=0}^{\infty}(\nu+2k)\BesselJ{\nu+2k}^{2}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t\BesselJ{\nu-1}^{2}@{t}\diff{t} = 2\sum_{k=0}^{\infty}(\nu+2k)\BesselJ{\nu+2k}^{2}@{x}</syntaxhighlight> || <math>\realpart@@{\nu} > 0, \realpart@@{((\nu-1)+k+1)} > 0, \realpart@@{((\nu+2k)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(t*(BesselJ(nu - 1, t))^(2), t = 0..x) = 2*sum((nu + 2*k)*(BesselJ(nu + 2*k, x))^(2), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[t*(BesselJ[\[Nu]- 1, t])^(2), {t, 0, x}, GenerateConditions->None] == 2*Sum[(\[Nu]+ 2*k)*(BesselJ[\[Nu]+ 2*k, x])^(2), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 15] || Successful [Tested: 15]
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 | [https://dlmf.nist.gov/10.22.E28 10.22.E28] || [[Item:Q3402|<math>\int_{0}^{x}t\left(\BesselJ{\nu-1}^{2}@{t}-\BesselJ{\nu+1}^{2}@{t}\right)\diff{t} = 2\nu\BesselJ{\nu}^{2}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t\left(\BesselJ{\nu-1}^{2}@{t}-\BesselJ{\nu+1}^{2}@{t}\right)\diff{t} = 2\nu\BesselJ{\nu}^{2}@{x}</syntaxhighlight> || <math>\realpart@@{\nu} > 0, \realpart@@{((\nu-1)+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(t*((BesselJ(nu - 1, t))^(2)- (BesselJ(nu + 1, t))^(2)), t = 0..x) = 2*nu*(BesselJ(nu, x))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[t*((BesselJ[\[Nu]- 1, t])^(2)- (BesselJ[\[Nu]+ 1, t])^(2)), {t, 0, x}, GenerateConditions->None] == 2*\[Nu]*(BesselJ[\[Nu], x])^(2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 15]
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 | [https://dlmf.nist.gov/10.22.E29 10.22.E29] || [[Item:Q3403|<math>\int_{0}^{x}t\BesselJ{0}^{2}@{t}\diff{t} = \tfrac{1}{2}x^{2}\left(\BesselJ{0}^{2}@{x}+\BesselJ{1}^{2}@{x}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t\BesselJ{0}^{2}@{t}\diff{t} = \tfrac{1}{2}x^{2}\left(\BesselJ{0}^{2}@{x}+\BesselJ{1}^{2}@{x}\right)</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(t*(BesselJ(0, t))^(2), t = 0..x) = (1)/(2)*(x)^(2)*((BesselJ(0, x))^(2)+ (BesselJ(1, x))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[t*(BesselJ[0, t])^(2), {t, 0, x}, GenerateConditions->None] == Divide[1,2]*(x)^(2)*((BesselJ[0, x])^(2)+ (BesselJ[1, x])^(2))</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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 | [https://dlmf.nist.gov/10.22.E30 10.22.E30] || [[Item:Q3404|<math>\int_{0}^{x}\BesselJ{n}@{t}\BesselJ{n+1}@{t}\diff{t} = \tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{n}@{t}\BesselJ{n+1}@{t}\diff{t} = \tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0, \realpart@@{((n+1)+k+1)} > 0, \realpart@@{(0+k+1)} > 0, \realpart@@{(k+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(n, t)*BesselJ(n + 1, t), t = 0..x) = (1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[n, t]*BesselJ[n + 1, t], {t, 0, x}, GenerateConditions->None] == Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.6308420033135872, DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[Plus[2, ], Power[1.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[1.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[1.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 1.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2], Times[Power[1.5, -2], Power[Plus[Times[-1, 1.5, BesselJ[0, 1.5]], Times[2, BesselJ[1, 1.5]]], 2]]]]}]][4.0]], {Rule[n, 3], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.9403627636501156, DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[Plus[2, ], Power[0.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[0.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[0.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[0.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[0.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 0.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2], Times[Power[0.5, -2], Power[Plus[Times[-1, 0.5, BesselJ[0, 0.5]], Times[2, BesselJ[1, 0.5]]], 2]]]]}]][4.0]], {Rule[n, 3], Rule[x, 0.5]}</syntaxhighlight><br></div></div>
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 | [https://dlmf.nist.gov/10.22.E30 10.22.E30] || [[Item:Q3404|<math>\tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x} = \sum_{k=n+1}^{\infty}\BesselJ{k}^{2}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}\left(1-\BesselJ{0}^{2}@{x}\right)-\sum_{k=1}^{n}\BesselJ{k}^{2}@{x} = \sum_{k=n+1}^{\infty}\BesselJ{k}^{2}@{x}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0, \realpart@@{((n+1)+k+1)} > 0, \realpart@@{(0+k+1)} > 0, \realpart@@{(k+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n) = sum((BesselJ(k, x))^(2), k = n + 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}, GenerateConditions->None] == Sum[(BesselJ[k, x])^(2), {k, n + 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.6309837827773054, Times[-1.0, NSum[Power[BesselJ[k, 1.5], 2]
 Test Values: {k, 4, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], Power[1.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[1.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[1.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 1.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 1.5], 2], Power[BesselJ[1, 1.5], 2], Times[Power[1.5, -2], Power[Plus[Times[-1, 1.5, BesselJ[0, 1.5]], Times[2, BesselJ[1, 1.5]]], 2]]]]}]][4.0]]], {Ru<syntaxhighlight lang=mathematica>Result: Plus[0.9403627895513045, Times[-1.0, NSum[Power[BesselJ[k, 0.5], 2]
 Test Values: {k, 4, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], Power[0.5, 2], []], Times[Plus[-8, Times[-20, ], Times[-16, Power[, 2]], Times[-4, Power[, 3]], Times[-1, Power[0.5, 2]]], [Plus[1, ]]], Times[Plus[3, Times[2, ]], Plus[8, Times[12, ], Times[4, Power[, 2]], Times[-1, Power[0.5, 2]]], [Plus[2, ]]], Times[Plus[-16, Times[-32, ], Times[-20, Power[, 2]], Times[-4, Power[, 3]], Power[0.5, 2]], [Plus[3, ]]], Times[Plus[1, ], Power[0.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Power[BesselJ[0, 0.5], 2]], Equal[[2], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2]]], Equal[[3], Plus[Power[BesselJ[0, 0.5], 2], Power[BesselJ[1, 0.5], 2], Times[Power[0.5, -2], Power[Plus[Times[-1, 0.5, BesselJ[0, 0.5]], Times[2, BesselJ[1, 0.5]]], 2]]]]}]][4.0]]], {Rule[n, 3], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
 | [https://dlmf.nist.gov/10.22.E31 10.22.E31] || [[Item:Q3405|<math>\int_{0}^{x}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = 2\sum_{k=0}^{\infty}(-1)^{k}\BesselJ{\mu+\nu+2k+1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = 2\sum_{k=0}^{\infty}(-1)^{k}\BesselJ{\mu+\nu+2k+1}@{x}</syntaxhighlight> || <math>\realpart@@{\mu} > -1, \realpart@@{\nu} > -1, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu+2k+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(mu, t)*BesselJ(nu, x - t), t = 0..x) = 2*sum((- 1)^(k)* BesselJ(mu + nu + 2*k + 1, x), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t], {t, 0, x}, GenerateConditions->None] == 2*Sum[(- 1)^(k)* BesselJ[\[Mu]+ \[Nu]+ 2*k + 1, x], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Skip - No test values generated
 |-
 | [https://dlmf.nist.gov/10.22.E32 10.22.E32] || [[Item:Q3406|<math>\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{1-\nu}@{x-t}\diff{t} = \BesselJ{0}@{x}-\cos@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{1-\nu}@{x-t}\diff{t} = \BesselJ{0}@{x}-\cos@@{x}</syntaxhighlight> || <math>-1 < \realpart@@{\nu}, \realpart@@{\nu} < 2, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((1-\nu)+k+1)} > 0, \realpart@@{(0+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(nu, t)*BesselJ(1 - nu, x - t), t = 0..x) = BesselJ(0, x)- cos(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Nu], t]*BesselJ[1 - \[Nu], x - t], {t, 0, x}, GenerateConditions->None] == BesselJ[0, x]- Cos[x]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
 |-
 | [https://dlmf.nist.gov/10.22.E33 10.22.E33] || [[Item:Q3407|<math>\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{-\nu}@{x-t}\diff{t} = \sin@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\BesselJ{\nu}@{t}\BesselJ{-\nu}@{x-t}\diff{t} = \sin@@{x}</syntaxhighlight> || <math>|\realpart@@{\nu}| < 1, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(BesselJ(nu, t)*BesselJ(- nu, x - t), t = 0..x) = sin(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BesselJ[\[Nu], t]*BesselJ[- \[Nu], x - t], {t, 0, x}, GenerateConditions->None] == Sin[x]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
 |-
 | [https://dlmf.nist.gov/10.22.E34 10.22.E34] || [[Item:Q3408|<math>\int_{0}^{x}t^{-1}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = \frac{\BesselJ{\mu+\nu}@{x}}{\mu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t^{-1}\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t} = \frac{\BesselJ{\mu+\nu}@{x}}{\mu}</syntaxhighlight> || <math>\realpart@@{\mu} > 0, \realpart@@{\nu} > -1, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(- 1)* BesselJ(mu, t)*BesselJ(nu, x - t), t = 0..x) = (BesselJ(mu + nu, x))/(mu)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(- 1)* BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t], {t, 0, x}, GenerateConditions->None] == Divide[BesselJ[\[Mu]+ \[Nu], x],\[Mu]]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skip - No test values generated
 |-
 | [https://dlmf.nist.gov/10.22.E35 10.22.E35] || [[Item:Q3409|<math>\int_{0}^{x}\frac{\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t}}{t(x-t)} = \frac{(\mu+\nu)\BesselJ{\mu+\nu}@{x}}{\mu\nu x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\frac{\BesselJ{\mu}@{t}\BesselJ{\nu}@{x-t}\diff{t}}{t(x-t)} = \frac{(\mu+\nu)\BesselJ{\mu+\nu}@{x}}{\mu\nu x}</syntaxhighlight> || <math>\realpart@@{\mu} > 0, \realpart@@{\nu} > 0, \realpart@@{((\mu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\mu+\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int((BesselJ(mu, t)*BesselJ(nu, x - t))/(t*(x - t)), t = 0..x) = ((mu + nu)*BesselJ(mu + nu, x))/(mu*nu*x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t],t*(x - t)], {t, 0, x}, GenerateConditions->None] == Divide[(\[Mu]+ \[Nu])*BesselJ[\[Mu]+ \[Nu], x],\[Mu]*\[Nu]*x]</syntaxhighlight> || Error || Failure || - || Skip - No test values generated
 |-
 | [https://dlmf.nist.gov/10.22.E36 10.22.E36] || [[Item:Q3410|<math>\frac{1}{\EulerGamma@{\alpha}}\int_{0}^{x}(x-t)^{\alpha-1}\BesselJ{\nu}@{t}\diff{t} = 2^{\alpha}\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{k!}\BesselJ{\nu+\alpha+2k}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{\alpha}}\int_{0}^{x}(x-t)^{\alpha-1}\BesselJ{\nu}@{t}\diff{t} = 2^{\alpha}\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{k!}\BesselJ{\nu+\alpha+2k}@{x}</syntaxhighlight> || <math>\realpart@@{\alpha} > 0, \realpart@@{\nu} \geq 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+\alpha+2k)+k+1)} > 0, \realpart@@{(\alpha)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(alpha))*int((x - t)^(alpha - 1)* BesselJ(nu, t), t = 0..x) = (2)^(alpha)* sum((alpha[k])/(factorial(k))*BesselJ(nu + alpha + 2*k, x), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[\[Alpha]]]*Integrate[(x - t)^(\[Alpha]- 1)* BesselJ[\[Nu], t], {t, 0, x}, GenerateConditions->None] == (2)^\[Alpha]* Sum[Divide[Subscript[\[Alpha], k],(k)!]*BesselJ[\[Nu]+ \[Alpha]+ 2*k, x], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Skip - No test values generated
 |-
 | [https://dlmf.nist.gov/10.22.E37 10.22.E37] || [[Item:Q3411|<math>\int_{0}^{1}t\BesselJ{\nu}@{j_{\nu,\ell}t}\BesselJ{\nu}@{j_{\nu,m}t}\diff{t} = \tfrac{1}{2}\left(\BesselJ{\nu}'@{j_{\nu,\ell}}\right)^{2}\Kroneckerdelta{\ell}{m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}t\BesselJ{\nu}@{j_{\nu,\ell}t}\BesselJ{\nu}@{j_{\nu,m}t}\diff{t} = \tfrac{1}{2}\left(\BesselJ{\nu}'@{j_{\nu,\ell}}\right)^{2}\Kroneckerdelta{\ell}{m}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(t*BesselJ(nu, j[nu , ell]*t)*BesselJ(nu, j[nu , m]*t), t = 0..1) = (1)/(2)*(diff( BesselJ(nu, j[nu , ell]), j[nu , ell]$(1) ))^(2)* KroneckerDelta[ell, m]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[t*BesselJ[\[Nu], Subscript[j, \[Nu], \[ScriptL]]*t]*BesselJ[\[Nu], Subscript[j, \[Nu], m]*t], {t, 0, 1}, GenerateConditions->None] == Divide[1,2]*(D[BesselJ[\[Nu], Subscript[j, \[Nu], \[ScriptL]]], {Subscript[j, \[Nu], \[ScriptL]], 1}])^(2)* KroneckerDelta[\[ScriptL], m]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
 Test Values: {Rule[m, 1], Rule[ℓ, 1], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, ν, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[j, ν, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate

Results of Bessel Functions I: Difference between revisions

Latest revision as of 06:59, 25 May 2021

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