Struve and Related Functions - 11.10 Anger–Weber Functions

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11.10.E1 𝐉 Ξ½ ⁑ ( z ) = 1 Ο€ ⁒ ∫ 0 Ο€ cos ⁑ ( Ξ½ ⁒ ΞΈ - z ⁒ sin ⁑ ΞΈ ) ⁒ d ΞΈ Anger-J 𝜈 𝑧 1 πœ‹ superscript subscript 0 πœ‹ 𝜈 πœƒ 𝑧 πœƒ πœƒ {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{% 0}^{\pi}\cos\left(\nu\theta-z\sin\theta\right)\mathrm{d}\theta}}
\AngerJ{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{\nu\theta-z\sin@@{\theta}}\diff{\theta}		 

AngerJ(nu, z) = (1)/(Pi)*int(cos(nu*theta - z*sin(theta)), theta = 0..Pi)

AngerJ[\[Nu], z] == Divide[1,Pi]*Integrate[Cos[\[Nu]*\[Theta]- z*Sin[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
Failure Aborted Successful [Tested: 70] Skipped - Because timed out
11.10.E2 𝐄 Ξ½ ⁑ ( z ) = 1 Ο€ ⁒ ∫ 0 Ο€ sin ⁑ ( Ξ½ ⁒ ΞΈ - z ⁒ sin ⁑ ΞΈ ) ⁒ d ΞΈ Weber-E 𝜈 𝑧 1 πœ‹ superscript subscript 0 πœ‹ 𝜈 πœƒ 𝑧 πœƒ πœƒ {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{% 0}^{\pi}\sin\left(\nu\theta-z\sin\theta\right)\mathrm{d}\theta}}
\WeberE{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\sin@{\nu\theta-z\sin@@{\theta}}\diff{\theta}		 

WeberE(nu, z) = (1)/(Pi)*int(sin(nu*theta - z*sin(theta)), theta = 0..Pi)

WeberE[\[Nu], z] == Divide[1,Pi]*Integrate[Sin[\[Nu]*\[Theta]- z*Sin[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
Failure Failure Successful [Tested: 70] Skipped - Because timed out
11.10.E3 1 Ο€ ⁒ ∫ 0 2 ⁒ Ο€ cos ⁑ ( Ξ½ ⁒ ΞΈ - z ⁒ sin ⁑ ΞΈ ) ⁒ d ΞΈ = ( 1 + cos ⁑ ( 2 ⁒ Ο€ ⁒ Ξ½ ) ) ⁒ 𝐉 Ξ½ ⁑ ( z ) + sin ⁑ ( 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ 𝐄 Ξ½ ⁑ ( z ) 1 πœ‹ superscript subscript 0 2 πœ‹ 𝜈 πœƒ 𝑧 πœƒ πœƒ 1 2 πœ‹ 𝜈 Anger-J 𝜈 𝑧 2 πœ‹ 𝜈 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{2\pi}\cos\left(\nu\theta-z% \sin\theta\right)\mathrm{d}\theta=(1+\cos\left(2\pi\nu\right))\,\mathbf{J}_{% \nu}\left(z\right)+\sin\left(2\pi\nu\right)\mathbf{E}_{\nu}\left(z\right)}}
\frac{1}{\pi}\int_{0}^{2\pi}\cos@{\nu\theta-z\sin@@{\theta}}\diff{\theta} = (1+\cos@{2\pi\nu})\,\AngerJ{\nu}@{z}+\sin@{2\pi\nu}\WeberE{\nu}@{z}		 

(1)/(Pi)*int(cos(nu*theta - z*sin(theta)), theta = 0..2*Pi) = (1 + cos(2*Pi*nu))*AngerJ(nu, z)+ sin(2*Pi*nu)*WeberE(nu, z)

Divide[1,Pi]*Integrate[Cos[\[Nu]*\[Theta]- z*Sin[\[Theta]]], {\[Theta], 0, 2*Pi}, GenerateConditions->None] == (1 + Cos[2*Pi*\[Nu]])*AngerJ[\[Nu], z]+ Sin[2*Pi*\[Nu]]*WeberE[\[Nu], z]
Failure Failure Successful [Tested: 70] Skipped - Because timed out
11.10.E8 𝐉 Ξ½ ⁑ ( z ) = cos ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 1 ⁒ ( Ξ½ , z ) + sin ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 2 ⁒ ( Ξ½ , z ) Anger-J 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 1 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 2 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(z\right)=\cos\left(\tfrac{1}% {2}\pi\nu\right)\,S_{1}(\nu,z)+\sin\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,% z)}}
\AngerJ{\nu}@{z} = \cos@{\tfrac{1}{2}\pi\nu}\,S_{1}(\nu,z)+\sin@{\tfrac{1}{2}\pi\nu}\,S_{2}(\nu,z)		 

AngerJ(nu, z) = cos((1)/(2)*Pi*nu)*S[1](nu , z)+ sin((1)/(2)*Pi*nu)*S[2](nu , z)

AngerJ[\[Nu], z] == Cos[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 1][\[Nu], z]+ Sin[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 2][\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: .4325617835-.4216939044*I-(1.695493166+.2075033380*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[1] = 1/2*3^(1/2)+1/2*I, S[2] = 1/2*3^(1/2)+1/2*I}


Result: .4325617835-.4216939044*I+(.1404557731-.4337646272*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[1] = 1/2*3^(1/2)+1/2*I, S[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
11.10.E9 𝐄 Ξ½ ⁑ ( z ) = sin ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 1 ⁒ ( Ξ½ , z ) - cos ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 2 ⁒ ( Ξ½ , z ) Weber-E 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 1 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 2 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(z\right)=\sin\left(\tfrac{1}% {2}\pi\nu\right)\,S_{1}(\nu,z)-\cos\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,% z)}}
\WeberE{\nu}@{z} = \sin@{\tfrac{1}{2}\pi\nu}\,S_{1}(\nu,z)-\cos@{\tfrac{1}{2}\pi\nu}\,S_{2}(\nu,z)		 

WeberE(nu, z) = sin((1)/(2)*Pi*nu)*S[1](nu , z)- cos((1)/(2)*Pi*nu)*S[2](nu , z)

WeberE[\[Nu], z] == Sin[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 1][\[Nu], z]- Cos[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 2][\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: .6530158617-.8867638354e-1*I-(.3667170623+1.402184312*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[1] = 1/2*3^(1/2)+1/2*I, S[2] = 1/2*3^(1/2)+1/2*I}


Result: .6530158617-.8867638354e-1*I-(.4337646272+.1404557731*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[1] = 1/2*3^(1/2)+1/2*I, S[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
11.10.E10 S 1 ⁒ ( Ξ½ , z ) = βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ ( 1 2 ⁒ z ) 2 ⁒ k Ξ“ ⁑ ( k + 1 2 ⁒ Ξ½ + 1 ) ⁒ Ξ“ ⁑ ( k - 1 2 ⁒ Ξ½ + 1 ) subscript 𝑆 1 𝜈 𝑧 superscript subscript π‘˜ 0 superscript 1 π‘˜ superscript 1 2 𝑧 2 π‘˜ Euler-Gamma π‘˜ 1 2 𝜈 1 Euler-Gamma π‘˜ 1 2 𝜈 1 {\displaystyle{\displaystyle S_{1}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(% \tfrac{1}{2}z)^{2k}}{\Gamma\left(k\!+\!\tfrac{1}{2}\nu+1\right)\Gamma\left(k\!% -\!\tfrac{1}{2}\nu\!+\!1\right)}}}
S_{1}(\nu,z) = \sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k}}{\EulerGamma@{k\!+\!\tfrac{1}{2}\nu+1}\EulerGamma@{k\!-\!\tfrac{1}{2}\nu\!+\!1}}		 β„œ ⁑ ( k + 1 2 ⁒ Ξ½ + 1 ) > 0 , β„œ ⁑ ( k - 1 2 ⁒ Ξ½ + 1 ) > 0 formulae-sequence π‘˜ 1 2 𝜈 1 0 π‘˜ 1 2 𝜈 1 0 {\displaystyle{\displaystyle\Re(k+\tfrac{1}{2}\nu+1)>0,\Re(k-\tfrac{1}{2}\nu+1% )>0}} 
S[1](nu , z) = sum(((- 1)^(k)*((1)/(2)*z)^(2*k))/(GAMMA(k +(1)/(2)*nu + 1)*GAMMA(k -(1)/(2)*nu + 1)), k = 0..infinity)

Subscript[S, 1][\[Nu], z] == Sum[Divide[(- 1)^(k)*(Divide[1,2]*z)^(2*k),Gamma[k +Divide[1,2]*\[Nu]+ 1]*Gamma[k -Divide[1,2]*\[Nu]+ 1]], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)-.6234597010+.4805214665*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[1] = 1/2*3^(1/2)+1/2*I}


Result: (-.5000000000+.8660254040*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)-.6234597010+.4805214665*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
11.10.E11 S 2 ⁒ ( Ξ½ , z ) = βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ ( 1 2 ⁒ z ) 2 ⁒ k + 1 Ξ“ ⁑ ( k + 1 2 ⁒ Ξ½ + 3 2 ) ⁒ Ξ“ ⁑ ( k - 1 2 ⁒ Ξ½ + 3 2 ) subscript 𝑆 2 𝜈 𝑧 superscript subscript π‘˜ 0 superscript 1 π‘˜ superscript 1 2 𝑧 2 π‘˜ 1 Euler-Gamma π‘˜ 1 2 𝜈 3 2 Euler-Gamma π‘˜ 1 2 𝜈 3 2 {\displaystyle{\displaystyle S_{2}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(% \tfrac{1}{2}z)^{2k+1}}{\Gamma\left(k\!+\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}% \right)\Gamma\left(k\!-\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}\right)}}}
S_{2}(\nu,z) = \sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k+1}}{\EulerGamma@{k\!+\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}}\EulerGamma@{k\!-\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}}}		 β„œ ⁑ ( k + 1 2 ⁒ Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( k - 1 2 ⁒ Ξ½ + 3 2 ) > 0 formulae-sequence π‘˜ 1 2 𝜈 3 2 0 π‘˜ 1 2 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(k+\tfrac{1}{2}\nu+\tfrac{3}{2})>0,\Re(k-\tfrac% {1}{2}\nu+\tfrac{3}{2})>0}} 
S[2](nu , z) = sum(((- 1)^(k)*((1)/(2)*z)^(2*k + 1))/(GAMMA(k +(1)/(2)*nu +(3)/(2))*GAMMA(k -(1)/(2)*nu +(3)/(2))), k = 0..infinity)

Subscript[S, 2][\[Nu], z] == Sum[Divide[(- 1)^(k)*(Divide[1,2]*z)^(2*k + 1),Gamma[k +Divide[1,2]*\[Nu]+Divide[3,2]]*Gamma[k -Divide[1,2]*\[Nu]+Divide[3,2]]], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)-.4892722811e-1+.1117224133*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[2] = 1/2*3^(1/2)+1/2*I}


Result: (-.5000000000+.8660254040*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)-.4892722811e-1+.1117224133*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, S[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
11.10#Ex1 𝐉 Ξ½ ⁑ ( - z ) = 𝐉 - Ξ½ ⁑ ( z ) Anger-J 𝜈 𝑧 Anger-J 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(-z\right)=\mathbf{J}_{-\nu}% \left(z\right)}}
\AngerJ{\nu}@{-z} = \AngerJ{-\nu}@{z}		 

AngerJ(nu, - z) = AngerJ(- nu, z)

AngerJ[\[Nu], - z] == AngerJ[- \[Nu], z]
Successful Successful - Successful [Tested: 70]
11.10#Ex2 𝐄 Ξ½ ⁑ ( - z ) = - 𝐄 - Ξ½ ⁑ ( z ) Weber-E 𝜈 𝑧 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(-z\right)=-\mathbf{E}_{-\nu}% \left(z\right)}}
\WeberE{\nu}@{-z} = -\WeberE{-\nu}@{z}		 

WeberE(nu, - z) = - WeberE(- nu, z)

WeberE[\[Nu], - z] == - WeberE[- \[Nu], z]
Successful Successful - Successful [Tested: 70]
11.10.E13 sin ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐉 Ξ½ ⁑ ( z ) = cos ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐄 Ξ½ ⁑ ( z ) - 𝐄 - Ξ½ ⁑ ( z ) πœ‹ 𝜈 Anger-J 𝜈 𝑧 πœ‹ 𝜈 Weber-E 𝜈 𝑧 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\pi\nu\right)\,\mathbf{J}_{\nu}\left(z% \right)=\cos\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z\right)-\mathbf{E}_{-% \nu}\left(z\right)}}
\sin@{\pi\nu}\,\AngerJ{\nu}@{z} = \cos@{\pi\nu}\,\WeberE{\nu}@{z}-\WeberE{-\nu}@{z}		 

sin(Pi*nu)*AngerJ(nu, z) = cos(Pi*nu)*WeberE(nu, z)- WeberE(- nu, z)

Sin[Pi*\[Nu]]*AngerJ[\[Nu], z] == Cos[Pi*\[Nu]]*WeberE[\[Nu], z]- WeberE[- \[Nu], z]
Successful Failure - Successful [Tested: 70]
11.10.E14 sin ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐄 Ξ½ ⁑ ( z ) = 𝐉 - Ξ½ ⁑ ( z ) - cos ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐉 Ξ½ ⁑ ( z ) πœ‹ 𝜈 Weber-E 𝜈 𝑧 Anger-J 𝜈 𝑧 πœ‹ 𝜈 Anger-J 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z% \right)=\mathbf{J}_{-\nu}\left(z\right)-\cos\left(\pi\nu\right)\,\mathbf{J}_{% \nu}\left(z\right)}}
\sin@{\pi\nu}\,\WeberE{\nu}@{z} = \AngerJ{-\nu}@{z}-\cos@{\pi\nu}\,\AngerJ{\nu}@{z}		 

sin(Pi*nu)*WeberE(nu, z) = AngerJ(- nu, z)- cos(Pi*nu)*AngerJ(nu, z)

Sin[Pi*\[Nu]]*WeberE[\[Nu], z] == AngerJ[- \[Nu], z]- Cos[Pi*\[Nu]]*AngerJ[\[Nu], z]
Successful Failure - Successful [Tested: 70]
11.10.E17 𝐉 Ξ½ ⁑ ( z ) = sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ ⁒ ( s 0 , Ξ½ ⁑ ( z ) - Ξ½ ⁒ s - 1 , Ξ½ ⁑ ( z ) ) Anger-J 𝜈 𝑧 πœ‹ 𝜈 πœ‹ Lommel-s 0 𝜈 𝑧 𝜈 Lommel-s 1 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(z\right)=\frac{\sin\left(\pi% \nu\right)}{\pi}(s_{{0},{\nu}}\left(z\right)-\nu s_{{-1},{\nu}}\left(z\right))}}
\AngerJ{\nu}@{z} = \frac{\sin@{\pi\nu}}{\pi}(\Lommels{0}{\nu}@{z}-\nu\Lommels{-1}{\nu}@{z})		 

AngerJ(nu, z) = (sin(Pi*nu))/(Pi)*(LommelS1(0, nu, z)- nu*LommelS1(- 1, nu, z))

Error
Successful Missing Macro Error - -
11.10.E18 𝐄 Ξ½ ⁑ ( z ) = - 1 Ο€ ⁒ ( 1 + cos ⁑ ( Ο€ ⁒ Ξ½ ) ) ⁒ s 0 , Ξ½ ⁑ ( z ) - Ξ½ Ο€ ⁒ ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) ⁒ s - 1 , Ξ½ ⁑ ( z ) Weber-E 𝜈 𝑧 1 πœ‹ 1 πœ‹ 𝜈 Lommel-s 0 𝜈 𝑧 𝜈 πœ‹ 1 πœ‹ 𝜈 Lommel-s 1 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(z\right)=-\frac{1}{\pi}(1+% \cos\left(\pi\nu\right))s_{{0},{\nu}}\left(z\right)\\ -\frac{\nu}{\pi}(1-\cos\left(\pi\nu\right))s_{{-1},{\nu}}\left(z\right)}}
\WeberE{\nu}@{z} = -\frac{1}{\pi}(1+\cos@{\pi\nu})\Lommels{0}{\nu}@{z}\\ -\frac{\nu}{\pi}(1-\cos@{\pi\nu})\Lommels{-1}{\nu}@{z}		 

WeberE(nu, z) = -(1)/(Pi)*(1 + cos(Pi*nu))*LommelS1(0, nu, z)*; -(nu)/(Pi)*(1 - cos(Pi*nu))* LommelS1(- 1, nu, z)

Error
Successful Missing Macro Error - -
11.10.E19 𝐉 - 1 2 ⁑ ( z ) = 𝐄 1 2 ⁑ ( z ) Anger-J 1 2 𝑧 Weber-E 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{-\frac{1}{2}}\left(z\right)=\mathbf{E}% _{\frac{1}{2}}\left(z\right)\\ }}
\AngerJ{-\frac{1}{2}}@{z} = \WeberE{\frac{1}{2}}@{z}\\		 

AngerJ(-(1)/(2), z) = WeberE((1)/(2), z)

AngerJ[-Divide[1,2], z] == WeberE[Divide[1,2], z]
Successful Successful - Successful [Tested: 7]
11.10.E19 𝐄 1 2 ⁑ ( z ) = ( 1 2 ⁒ Ο€ ⁒ z ) - 1 2 ⁒ ( A + ⁒ ( Ο‡ ) ⁒ cos ⁑ z - A - ⁒ ( Ο‡ ) ⁒ sin ⁑ z ) Weber-E 1 2 𝑧 superscript 1 2 πœ‹ 𝑧 1 2 subscript 𝐴 πœ’ 𝑧 subscript 𝐴 πœ’ 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\frac{1}{2}}\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\cos z-A_{-}(\chi)\sin z)}}
\WeberE{\frac{1}{2}}@{z}\\ = (\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\cos@@{z}-A_{-}(\chi)\sin@@{z})		 

WeberE((1)/(2), z) = ((1)/(2)*Pi*z)^(-(1)/(2))*(A[+](chi)* cos(z)- A[-](chi)* sin(z))

WeberE[Divide[1,2], z] == (Divide[1,2]*Pi*z)^(-Divide[1,2])*(Subscript[A, +][\[Chi]]* Cos[z]- Subscript[A, -][\[Chi]]* Sin[z])
Error Failure - Error
11.10.E20 𝐉 1 2 ⁑ ( z ) = - 𝐄 - 1 2 ⁑ ( z ) Anger-J 1 2 𝑧 Weber-E 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\frac{1}{2}}\left(z\right)=-\mathbf{E}% _{-\frac{1}{2}}\left(z\right)\\ }}
\AngerJ{\frac{1}{2}}@{z} = -\WeberE{-\frac{1}{2}}@{z}\\		 

AngerJ((1)/(2), z) = - WeberE(-(1)/(2), z)

AngerJ[Divide[1,2], z] == - WeberE[-Divide[1,2], z]
Successful Successful - Successful [Tested: 7]
11.10.E20 - 𝐄 - 1 2 ⁑ ( z ) = ( 1 2 ⁒ Ο€ ⁒ z ) - 1 2 ⁒ ( A + ⁒ ( Ο‡ ) ⁒ sin ⁑ z + A - ⁒ ( Ο‡ ) ⁒ cos ⁑ z ) Weber-E 1 2 𝑧 superscript 1 2 πœ‹ 𝑧 1 2 subscript 𝐴 πœ’ 𝑧 subscript 𝐴 πœ’ 𝑧 {\displaystyle{\displaystyle-\mathbf{E}_{-\frac{1}{2}}\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\sin z+A_{-}(\chi)\cos z)}}
-\WeberE{-\frac{1}{2}}@{z}\\ = (\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\sin@@{z}+A_{-}(\chi)\cos@@{z})		 

- WeberE(-(1)/(2), z) = ((1)/(2)*Pi*z)^(-(1)/(2))*(A[+](chi)* sin(z)+ A[-](chi)* cos(z))

- WeberE[-Divide[1,2], z] == (Divide[1,2]*Pi*z)^(-Divide[1,2])*(Subscript[A, +][\[Chi]]* Sin[z]+ Subscript[A, -][\[Chi]]* Cos[z])
Error Failure - Error
11.10#Ex3 A + ⁒ ( Ο‡ ) = C ⁑ ( Ο‡ ) + S ⁑ ( Ο‡ ) subscript 𝐴 πœ’ Fresnel-cosine-integral πœ’ Fresnel-sine-integral πœ’ {\displaystyle{\displaystyle A_{+}(\chi)=C\left(\chi\right)+S\left(\chi\right)}}
A_{+}(\chi) = \Fresnelcosint@{\chi}+\Fresnelsinint@{\chi}		 

A[+](chi) = FresnelC(chi)+ FresnelS(chi)

Subscript[A, +][\[Chi]] == FresnelC[\[Chi]]+ FresnelS[\[Chi]]
Error Failure - Error
11.10#Ex3 A - ⁒ ( Ο‡ ) = C ⁑ ( Ο‡ ) - S ⁑ ( Ο‡ ) subscript 𝐴 πœ’ Fresnel-cosine-integral πœ’ Fresnel-sine-integral πœ’ {\displaystyle{\displaystyle A_{-}(\chi)=C\left(\chi\right)-S\left(\chi\right)}}
A_{-}(\chi) = \Fresnelcosint@{\chi}-\Fresnelsinint@{\chi}		 

A[-](chi) = FresnelC(chi)- FresnelS(chi)

Subscript[A, -][\[Chi]] == FresnelC[\[Chi]]- FresnelS[\[Chi]]
Error Failure - Error
11.10#Ex4 Ο‡ = ( 2 ⁒ z / Ο€ ) 1 2 πœ’ superscript 2 𝑧 πœ‹ 1 2 {\displaystyle{\displaystyle\chi=(2z/\pi)^{\frac{1}{2}}}}
\chi = (2z/\pi)^{\frac{1}{2}}		 

chi = (2*z/Pi)^((1)/(2))
 
\[Chi] == (2*z/Pi)^(Divide[1,2])
 Skipped - no semantic math Skipped - no semantic math - -
11.10.E22 𝐄 n ⁑ ( z ) = - 𝐇 n ⁑ ( z ) + 1 Ο€ ⁒ βˆ‘ k = 0 m 1 Ξ“ ⁑ ( k + 1 2 ) Ξ“ ⁑ ( n + 1 2 - k ) ⁒ ( 1 2 ⁒ z ) n - 2 ⁒ k - 1 Weber-E 𝑛 𝑧 Struve-H 𝑛 𝑧 1 πœ‹ superscript subscript π‘˜ 0 subscript π‘š 1 Euler-Gamma π‘˜ 1 2 Euler-Gamma 𝑛 1 2 π‘˜ superscript 1 2 𝑧 𝑛 2 π‘˜ 1 {\displaystyle{\displaystyle\mathbf{E}_{n}\left(z\right)=-\mathbf{H}_{n}\left(% z\right)+\frac{1}{\pi}\sum_{k=0}^{m_{1}}\frac{\Gamma\left(k+\tfrac{1}{2}\right% )}{\Gamma\left(n\!+\!\tfrac{1}{2}\!-\!k\right)}(\tfrac{1}{2}z)^{n-2k-1}}}
\WeberE{n}@{z} = -\StruveH{n}@{z}+\frac{1}{\pi}\sum_{k=0}^{m_{1}}\frac{\EulerGamma@{k+\tfrac{1}{2}}}{\EulerGamma@{n\!+\!\tfrac{1}{2}\!-\!k}}(\tfrac{1}{2}z)^{n-2k-1}		 β„œ ⁑ ( k + 1 2 ) > 0 , β„œ ⁑ ( n + 1 2 - k ) > 0 , β„œ ⁑ ( n + n + 3 2 ) > 0 formulae-sequence π‘˜ 1 2 0 formulae-sequence 𝑛 1 2 π‘˜ 0 𝑛 𝑛 3 2 0 {\displaystyle{\displaystyle\Re(k+\tfrac{1}{2})>0,\Re(n+\tfrac{1}{2}-k)>0,\Re(% n+n+\tfrac{3}{2})>0}} 
WeberE(n, z) = - StruveH(n, z)+(1)/(Pi)*sum((GAMMA(k +(1)/(2)))/(GAMMA(n +(1)/(2)- k))*((1)/(2)*z)^(n - 2*k - 1), k = 0..m[1])

WeberE[n, z] == - StruveH[n, z]+Divide[1,Pi]*Sum[Divide[Gamma[k +Divide[1,2]],Gamma[n +Divide[1,2]- k]]*(Divide[1,2]*z)^(n - 2*k - 1), {k, 0, Subscript[m, 1]}, GenerateConditions->None]
Failure Failure Manual Skip!
Failed [210 / 210]
Result: Plus[0.6366197723675814, Times[-0.3183098861837907, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[2, ]], Plus[1, Times[2, ]], []], Times[Plus[-1, Times[-1, Power[-1, Rational[1, 3]]], Times[4, Power[, 2]]], [Plus[1, ]]], Times[Power[-1, Rational[1, 3]], [Plus[2, ]]]], 0], Equal[[0], 0], Equal[[1], 2]}]][Complex[1.8660254037844388, 0.49999999999999994]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.18377629847393068, 0.10610329539459687], Times[Complex[-0.13783222385544802, -0.07957747154594766], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Plus[-3, Times[2, ]], Plus[1, Times[2, ]], []], Times[Plus[-3, Times[-1, Power[-1, Rational[1, 3]]], Times[-4, ], Times[4, Power[, 2]]], [Plus[1, ]]], Times[Power[-1, Rational[1, 3]], [Plus[2, ]]]], 0], Equal[[0], 0], Equal[[1], Rational[4, 3]]}]][Complex[1.8660254037844388, 0.49999999999999994]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
11.10.E23 𝐄 - n ⁑ ( z ) = - 𝐇 - n ⁑ ( z ) + ( - 1 ) n + 1 Ο€ ⁒ βˆ‘ k = 0 m 2 Ξ“ ⁑ ( n - k - 1 2 ) Ξ“ ⁑ ( k + 3 2 ) ⁒ ( 1 2 ⁒ z ) - n + 2 ⁒ k + 1 Weber-E 𝑛 𝑧 Struve-H 𝑛 𝑧 superscript 1 𝑛 1 πœ‹ superscript subscript π‘˜ 0 subscript π‘š 2 Euler-Gamma 𝑛 π‘˜ 1 2 Euler-Gamma π‘˜ 3 2 superscript 1 2 𝑧 𝑛 2 π‘˜ 1 {\displaystyle{\displaystyle\mathbf{E}_{-n}\left(z\right)=-\mathbf{H}_{-n}% \left(z\right)+\frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{m_{2}}\frac{\Gamma\left(n\!-% \!k\!-\!\tfrac{1}{2}\right)}{\Gamma\left(k+\tfrac{3}{2}\right)}(\tfrac{1}{2}z)% ^{-n+2k+1}}}
\WeberE{-n}@{z} = -\StruveH{-n}@{z}+\frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{m_{2}}\frac{\EulerGamma@{n\!-\!k\!-\!\tfrac{1}{2}}}{\EulerGamma@{k+\tfrac{3}{2}}}(\tfrac{1}{2}z)^{-n+2k+1}		 β„œ ⁑ ( n - k - 1 2 ) > 0 , β„œ ⁑ ( k + 3 2 ) > 0 , β„œ ⁑ ( n + ( - n ) + 3 2 ) > 0 formulae-sequence 𝑛 π‘˜ 1 2 0 formulae-sequence π‘˜ 3 2 0 𝑛 𝑛 3 2 0 {\displaystyle{\displaystyle\Re(n-k-\tfrac{1}{2})>0,\Re(k+\tfrac{3}{2})>0,\Re(% n+(-n)+\tfrac{3}{2})>0}} 
WeberE(- n, z) = - StruveH(- n, z)+((- 1)^(n + 1))/(Pi)*sum((GAMMA(n - k -(1)/(2)))/(GAMMA(k +(3)/(2)))*((1)/(2)*z)^(- n + 2*k + 1), k = 0..m[2])

WeberE[- n, z] == - StruveH[- n, z]+Divide[(- 1)^(n + 1),Pi]*Sum[Divide[Gamma[n - k -Divide[1,2]],Gamma[k +Divide[3,2]]]*(Divide[1,2]*z)^(- n + 2*k + 1), {k, 0, Subscript[m, 2]}, GenerateConditions->None]
Failure Failure
Failed [210 / 210]
Result: -.5182370935+.1715162156*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m[2] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .1977910573+.6179671328e-1*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m[2] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[-0.5182370936641069, 0.17151621559870867]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.19779105745155356, 0.06179671324201291]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[m, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
11.10#Ex5 m 1 = ⌊ 1 2 ⁒ n - 1 2 βŒ‹ subscript π‘š 1 1 2 𝑛 1 2 {\displaystyle{\displaystyle m_{1}=\left\lfloor\tfrac{1}{2}n-\tfrac{1}{2}% \right\rfloor}}
m_{1} = \floor{\tfrac{1}{2}n-\tfrac{1}{2}}		 

m[1] = floor((1)/(2)*n -(1)/(2))

Subscript[m, 1] == Floor[Divide[1,2]*n -Divide[1,2]]
Failure Failure
Failed [30 / 30]
Result: .8660254040+.5000000000*I
Test Values: {m[1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .8660254040+.5000000000*I
Test Values: {m[1] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[Subscript[m, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[Subscript[m, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
11.10#Ex6 m 2 = ⌈ 1 2 ⁒ n - 3 2 βŒ‰ subscript π‘š 2 1 2 𝑛 3 2 {\displaystyle{\displaystyle m_{2}=\left\lceil\tfrac{1}{2}n-\tfrac{3}{2}\right% \rceil}}
m_{2} = \ceiling{\tfrac{1}{2}n-\tfrac{3}{2}}		 

m[2] = ceil((1)/(2)*n -(3)/(2))

Subscript[m, 2] == Ceiling[Divide[1,2]*n -Divide[3,2]]
Failure Failure
Failed [30 / 30]
Result: 1.866025404+.5000000000*I
Test Values: {m[2] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .8660254040+.5000000000*I
Test Values: {m[2] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[Subscript[m, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[Subscript[m, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
11.10.E25 𝐉 Ξ½ ⁑ ( 0 ) = sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ ⁒ Ξ½ Anger-J 𝜈 0 πœ‹ 𝜈 πœ‹ 𝜈 {\displaystyle{\displaystyle\displaystyle\mathbf{J}_{\nu}\left(0\right)=\frac{% \sin\left(\pi\nu\right)}{\pi\nu}}}
\displaystyle\AngerJ{\nu}@{0} = \frac{\sin@{\pi\nu}}{\pi\nu}		 

AngerJ(nu, 0) = (sin(Pi*nu))/(Pi*nu)
 
AngerJ[\[Nu], 0] == Divide[Sin[Pi*\[Nu]],Pi*\[Nu]]
 Skipped - no semantic math Skipped - no semantic math - -
11.10.E25 𝐄 Ξ½ ⁑ ( 0 ) = 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ ⁒ Ξ½ Weber-E 𝜈 0 1 πœ‹ 𝜈 πœ‹ 𝜈 {\displaystyle{\displaystyle\displaystyle\mathbf{E}_{\nu}\left(0\right)=\frac{% 1-\cos\left(\pi\nu\right)}{\pi\nu}}}
\displaystyle\WeberE{\nu}@{0} = \frac{1-\cos@{\pi\nu}}{\pi\nu}		 

WeberE(nu, 0) = (1 - cos(Pi*nu))/(Pi*nu)
 
WeberE[\[Nu], 0] == Divide[1 - Cos[Pi*\[Nu]],Pi*\[Nu]]
 Skipped - no semantic math Skipped - no semantic math - -
11.10.E26 𝐄 0 ⁑ ( z ) = - 𝐇 0 ⁑ ( z ) Weber-E 0 𝑧 Struve-H 0 𝑧 {\displaystyle{\displaystyle\displaystyle\mathbf{E}_{0}\left(z\right)=-\mathbf% {H}_{0}\left(z\right)}}
\displaystyle\WeberE{0}@{z} = -\StruveH{0}@{z}		 β„œ ⁑ ( n + 0 + 3 2 ) > 0 𝑛 0 3 2 0 {\displaystyle{\displaystyle\Re(n+0+\tfrac{3}{2})>0}} 
WeberE(0, z) = - StruveH(0, z)
 
WeberE[0, z] == - StruveH[0, z]
 Skipped - no semantic math Skipped - no semantic math - -
11.10.E26 𝐄 1 ⁑ ( z ) = 2 Ο€ - 𝐇 1 ⁑ ( z ) Weber-E 1 𝑧 2 πœ‹ Struve-H 1 𝑧 {\displaystyle{\displaystyle\displaystyle\mathbf{E}_{1}\left(z\right)=\frac{2}% {\pi}-\mathbf{H}_{1}\left(z\right)}}
\displaystyle\WeberE{1}@{z} = \frac{2}{\pi}-\StruveH{1}@{z}		 β„œ ⁑ ( n + 1 + 3 2 ) > 0 𝑛 1 3 2 0 {\displaystyle{\displaystyle\Re(n+1+\tfrac{3}{2})>0}} 
WeberE(1, z) = (2)/(Pi)- StruveH(1, z)
 
WeberE[1, z] == Divide[2,Pi]- StruveH[1, z]
 Skipped - no semantic math Skipped - no semantic math - -
11.10.E29 𝐉 n ⁑ ( z ) = J n ⁑ ( z ) Anger-J 𝑛 𝑧 Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{n}\left(z\right)=J_{n}\left(z\right)}}
\AngerJ{n}@{z} = \BesselJ{n}@{z}		 β„œ ⁑ ( n + k + 1 ) > 0 𝑛 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
AngerJ(n, z) = BesselJ(n, z)

AngerJ[n, z] == BesselJ[n, z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
11.10.E32 𝐉 Ξ½ - 1 ⁑ ( z ) + 𝐉 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ Ξ½ z ⁒ 𝐉 Ξ½ ⁑ ( z ) - 2 Ο€ ⁒ z ⁒ sin ⁑ ( Ο€ ⁒ Ξ½ ) Anger-J 𝜈 1 𝑧 Anger-J 𝜈 1 𝑧 2 𝜈 𝑧 Anger-J 𝜈 𝑧 2 πœ‹ 𝑧 πœ‹ 𝜈 {\displaystyle{\displaystyle\mathbf{J}_{\nu-1}\left(z\right)+\mathbf{J}_{\nu+1% }\left(z\right)=\frac{2\nu}{z}\mathbf{J}_{\nu}\left(z\right)-\frac{2}{\pi z}% \sin\left(\pi\nu\right)}}
\AngerJ{\nu-1}@{z}+\AngerJ{\nu+1}@{z} = \frac{2\nu}{z}\AngerJ{\nu}@{z}-\frac{2}{\pi z}\sin@{\pi\nu}		 

AngerJ(nu - 1, z)+ AngerJ(nu + 1, z) = (2*nu)/(z)*AngerJ(nu, z)-(2)/(Pi*z)*sin(Pi*nu)
 
AngerJ[\[Nu]- 1, z]+ AngerJ[\[Nu]+ 1, z] == Divide[2*\[Nu],z]*AngerJ[\[Nu], z]-Divide[2,Pi*z]*Sin[Pi*\[Nu]]
 Failure Failure
Failed [3 / 70]
Result: .1812319651
Test Values: {nu = -3/2, z = 3/2}

Result: .1208213102
Test Values: {nu = -1/2, z = 3/2}

... skip entries to safe data
 Successful [Tested: 70]
11.10.E33 𝐄 Ξ½ - 1 ⁑ ( z ) + 𝐄 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ Ξ½ z ⁒ 𝐄 Ξ½ ⁑ ( z ) - 2 Ο€ ⁒ z ⁒ ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) Weber-E 𝜈 1 𝑧 Weber-E 𝜈 1 𝑧 2 𝜈 𝑧 Weber-E 𝜈 𝑧 2 πœ‹ 𝑧 1 πœ‹ 𝜈 {\displaystyle{\displaystyle\mathbf{E}_{\nu-1}\left(z\right)+\mathbf{E}_{\nu+1% }\left(z\right)=\frac{2\nu}{z}\mathbf{E}_{\nu}\left(z\right)-\frac{2}{\pi z}(1% -\cos\left(\pi\nu\right))}}
\WeberE{\nu-1}@{z}+\WeberE{\nu+1}@{z} = \frac{2\nu}{z}\WeberE{\nu}@{z}-\frac{2}{\pi z}(1-\cos@{\pi\nu})		 

WeberE(nu - 1, z)+ WeberE(nu + 1, z) = (2*nu)/(z)*WeberE(nu, z)-(2)/(Pi*z)*(1 - cos(Pi*nu))
 
WeberE[\[Nu]- 1, z]+ WeberE[\[Nu]+ 1, z] == Divide[2*\[Nu],z]*WeberE[\[Nu], z]-Divide[2,Pi*z]*(1 - Cos[Pi*\[Nu]])
 Failure Failure
Failed [3 / 70]
Result: .1812319648
Test Values: {nu = 3/2, z = 3/2}

Result: .1812319652
Test Values: {nu = -1/2, z = 3/2}

... skip entries to safe data
 Successful [Tested: 70]
11.10.E34 2 ⁒ 𝐉 Ξ½ β€² ⁑ ( z ) = 𝐉 Ξ½ - 1 ⁑ ( z ) - 𝐉 Ξ½ + 1 ⁑ ( z ) 2 diffop Anger-J 𝜈 1 𝑧 Anger-J 𝜈 1 𝑧 Anger-J 𝜈 1 𝑧 {\displaystyle{\displaystyle 2\mathbf{J}_{\nu}'\left(z\right)=\mathbf{J}_{\nu-% 1}\left(z\right)-\mathbf{J}_{\nu+1}\left(z\right)}}
2\AngerJ{\nu}'@{z} = \AngerJ{\nu-1}@{z}-\AngerJ{\nu+1}@{z}		 

2*diff( AngerJ(nu, z), z$(1) ) = AngerJ(nu - 1, z)- AngerJ(nu + 1, z)
 
2*D[AngerJ[\[Nu], z], {z, 1}] == AngerJ[\[Nu]- 1, z]- AngerJ[\[Nu]+ 1, z]
 Failure Successful
Failed [3 / 70]
Result: -.1812319651
Test Values: {nu = -3/2, z = 3/2}

Result: -.1208213102
Test Values: {nu = -1/2, z = 3/2}

... skip entries to safe data
 Successful [Tested: 70]
11.10.E35 2 ⁒ 𝐄 Ξ½ β€² ⁑ ( z ) = 𝐄 Ξ½ - 1 ⁑ ( z ) - 𝐄 Ξ½ + 1 ⁑ ( z ) 2 diffop Weber-E 𝜈 1 𝑧 Weber-E 𝜈 1 𝑧 Weber-E 𝜈 1 𝑧 {\displaystyle{\displaystyle 2\mathbf{E}_{\nu}'\left(z\right)=\mathbf{E}_{\nu-% 1}\left(z\right)-\mathbf{E}_{\nu+1}\left(z\right)}}
2\WeberE{\nu}'@{z} = \WeberE{\nu-1}@{z}-\WeberE{\nu+1}@{z}		 

2*diff( WeberE(nu, z), z$(1) ) = WeberE(nu - 1, z)- WeberE(nu + 1, z)
 
2*D[WeberE[\[Nu], z], {z, 1}] == WeberE[\[Nu]- 1, z]- WeberE[\[Nu]+ 1, z]
 Failure Successful
Failed [3 / 70]
Result: -.1812319648
Test Values: {nu = 3/2, z = 3/2}

Result: -.1812319652
Test Values: {nu = -1/2, z = 3/2}

... skip entries to safe data
 Successful [Tested: 70]
11.10.E36 z ⁒ 𝐉 Ξ½ β€² ⁑ ( z ) + Ξ½ ⁒ 𝐉 Ξ½ ⁑ ( z ) = + z ⁒ 𝐉 Ξ½ - 1 ⁑ ( z ) + sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ 𝑧 diffop Anger-J 𝜈 1 𝑧 𝜈 Anger-J 𝜈 𝑧 𝑧 Anger-J 𝜈 1 𝑧 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{J}_{\nu}'\left(z\right)+\nu\mathbf{J}_{% \nu}\left(z\right)=+z\mathbf{J}_{\nu-1}\left(z\right)+\frac{\sin\left(\pi\nu% \right)}{\pi}}}
z\AngerJ{\nu}'@{z}+\nu\AngerJ{\nu}@{z} = + z\AngerJ{\nu- 1}@{z}+\frac{\sin@{\pi\nu}}{\pi}		 

z*diff( AngerJ(nu, z), z$(1) )+ nu*AngerJ(nu, z) = + z*AngerJ(nu - 1, z)+(sin(Pi*nu))/(Pi)
 
z*D[AngerJ[\[Nu], z], {z, 1}]+ \[Nu]*AngerJ[\[Nu], z] == + z*AngerJ[\[Nu]- 1, z]+Divide[Sin[Pi*\[Nu]],Pi]
 Failure Failure
Failed [3 / 70]
Result: -.2718479477
Test Values: {nu = -3/2, z = 3/2}

Result: -.1812319655
Test Values: {nu = -1/2, z = 3/2}

... skip entries to safe data
 Successful [Tested: 70]
11.10.E36 z ⁒ 𝐉 Ξ½ β€² ⁑ ( z ) - Ξ½ ⁒ 𝐉 Ξ½ ⁑ ( z ) = - z ⁒ 𝐉 Ξ½ + 1 ⁑ ( z ) - sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ 𝑧 diffop Anger-J 𝜈 1 𝑧 𝜈 Anger-J 𝜈 𝑧 𝑧 Anger-J 𝜈 1 𝑧 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{J}_{\nu}'\left(z\right)-\nu\mathbf{J}_{% \nu}\left(z\right)=-z\mathbf{J}_{\nu+1}\left(z\right)-\frac{\sin\left(\pi\nu% \right)}{\pi}}}
z\AngerJ{\nu}'@{z}-\nu\AngerJ{\nu}@{z} = - z\AngerJ{\nu+ 1}@{z}-\frac{\sin@{\pi\nu}}{\pi}		 

z*diff( AngerJ(nu, z), z$(1) )- nu*AngerJ(nu, z) = - z*AngerJ(nu + 1, z)-(sin(Pi*nu))/(Pi)
 
z*D[AngerJ[\[Nu], z], {z, 1}]- \[Nu]*AngerJ[\[Nu], z] == - z*AngerJ[\[Nu]+ 1, z]-Divide[Sin[Pi*\[Nu]],Pi]
 Successful Failure Skip - symbolical successful subtest Successful [Tested: 70]
11.10.E37 z ⁒ 𝐄 Ξ½ β€² ⁑ ( z ) + Ξ½ ⁒ 𝐄 Ξ½ ⁑ ( z ) = + z ⁒ 𝐄 Ξ½ - 1 ⁑ ( z ) + ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) Ο€ 𝑧 diffop Weber-E 𝜈 1 𝑧 𝜈 Weber-E 𝜈 𝑧 𝑧 Weber-E 𝜈 1 𝑧 1 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{E}_{\nu}'\left(z\right)+\nu\mathbf{E}_{% \nu}\left(z\right)=+z\mathbf{E}_{\nu-1}\left(z\right)+\frac{(1-\cos\left(\pi% \nu\right))}{\pi}}}
z\WeberE{\nu}'@{z}+\nu\WeberE{\nu}@{z} = + z\WeberE{\nu- 1}@{z}+\frac{(1-\cos@{\pi\nu})}{\pi}		 

z*diff( WeberE(nu, z), z$(1) )+ nu*WeberE(nu, z) = + z*WeberE(nu - 1, z)+(1 - cos(Pi*nu))/(Pi)
 
z*D[WeberE[\[Nu], z], {z, 1}]+ \[Nu]*WeberE[\[Nu], z] == + z*WeberE[\[Nu]- 1, z]+Divide[1 - Cos[Pi*\[Nu]],Pi]
 Failure Failure
Failed [3 / 70]
Result: -.2718479477
Test Values: {nu = 3/2, z = 3/2}

Result: -.2718479472
Test Values: {nu = -1/2, z = 3/2}

... skip entries to safe data
 Successful [Tested: 70]
11.10.E37 z ⁒ 𝐄 Ξ½ β€² ⁑ ( z ) - Ξ½ ⁒ 𝐄 Ξ½ ⁑ ( z ) = - z ⁒ 𝐄 Ξ½ + 1 ⁑ ( z ) - ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) Ο€ 𝑧 diffop Weber-E 𝜈 1 𝑧 𝜈 Weber-E 𝜈 𝑧 𝑧 Weber-E 𝜈 1 𝑧 1 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{E}_{\nu}'\left(z\right)-\nu\mathbf{E}_{% \nu}\left(z\right)=-z\mathbf{E}_{\nu+1}\left(z\right)-\frac{(1-\cos\left(\pi% \nu\right))}{\pi}}}
z\WeberE{\nu}'@{z}-\nu\WeberE{\nu}@{z} = - z\WeberE{\nu+ 1}@{z}-\frac{(1-\cos@{\pi\nu})}{\pi}		 

z*diff( WeberE(nu, z), z$(1) )- nu*WeberE(nu, z) = - z*WeberE(nu + 1, z)-(1 - cos(Pi*nu))/(Pi)
 
z*D[WeberE[\[Nu], z], {z, 1}]- \[Nu]*WeberE[\[Nu], z] == - z*WeberE[\[Nu]+ 1, z]-Divide[1 - Cos[Pi*\[Nu]],Pi]
 Successful Failure Skip - symbolical successful subtest Successful [Tested: 70]
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