10.24

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.24.E1	${\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d}x}+(x^{2}+\nu^{2})w=0}}$ x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0		(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))*w = 0	(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.948557159+2.125000000*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: .2165063513+1.125000001*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.9485571585149875, 2.125] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.948557158514987, 0.12499999999999989] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.24#Ex1	${\displaystyle{\displaystyle\widetilde{J}_{\nu}\left(x\right)=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(J_{i\nu}\left(x\right)\right)}}$ \BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}	${\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0}}$	sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))	Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]	Successful	Successful	-	Successful [Tested: 30]
10.24#Ex2	${\displaystyle{\displaystyle\widetilde{Y}_{\nu}\left(x\right)=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(Y_{i\nu}\left(x\right)\right)}}$ \BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}	${\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+k+1)>0}}$	sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))	Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]	Successful	Successful	-	Successful [Tested: 30]
10.24.E3	${\displaystyle{\displaystyle\Gamma\left(1+i\nu\right)=\left(\frac{\pi\nu}{\sinh\left(\pi\nu\right)}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}}}$ \EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}	${\displaystyle{\displaystyle\Re(1+\mathrm{i}\nu)>0}}$	GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])	Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]	Failure	Failure	Failed [300 / 300] Result: .131682196e-1-.6479738907*I Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I} Result: .2393622021-.2867640040*I Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.013168219691258531, -0.6479738909120968] Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.23936220222535412, -0.28676400411697583] Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.24#Ex3	${\displaystyle{\displaystyle\widetilde{J}_{-\nu}\left(x\right)=\widetilde{J}_{\nu}\left(x\right)}}$ \BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}		sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))	Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]	Failure	Failure	Failed [12 / 30] Result: .1765981285-.1547836875*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -1.059084556+.9282601935*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.6353785354467336, 0.04153700144653363] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.2910880978413849, 0.681683596996288] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.24#Ex4	${\displaystyle{\displaystyle\widetilde{Y}_{-\nu}\left(x\right)=\widetilde{Y}_{\nu}\left(x\right)}}$ \BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}	${\displaystyle{\displaystyle\Re((\mathrm{i}(-\nu))+k+1)>0,\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}(-\nu)))+k+1)>0,\Re((-(\mathrm{i}\nu))+k+1)>0}}$	sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))	Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]	Failure	Failure	Failed [12 / 30] Result: -.6730010946+.5898680353*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.1980888923+.1736197856*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[0.16541121369118172, 0.7534126929509344] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.3242468905843751, -0.9796849117084342] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.24.E5	${\displaystyle{\displaystyle\mathscr{W}\left\{\widetilde{J}_{\nu}\left(x\right),\widetilde{Y}_{\nu}\left(x\right)\right\}=2/(\pi x)}}$ \Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)	${\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+k+1)>0}}$	(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)	Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)	Failure	Failure	Failed [12 / 30] Result: -.3214564733-.7786157192*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.6431025084-4.765445687*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [30 / 30] Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-0.4244131815783876, Times[Complex[-0.5062208144169521, 0.3689208146583662], Plus[Times[Complex[1.2690034139339206, -1.428145592425075], Derivative[1][Re][Complex[-0.5230512553281585, -0.7250724679588263]]], Times[Complex[0.9907135967899046, 0.5862869255257461], Derivative[1][Re][Complex[0.9118063408652576, -0.381897212811936]]]]]] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.24.E9	${\displaystyle{\displaystyle\widetilde{Y}_{0}\left(x\right)=Y_{0}\left(x\right)}}$ \BesselYimag{0}@{x} = \BesselY{0}@{x}	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0,\Re((\mathrm{i}0)+k+1)>0,\Re((-(\mathrm{i}0))+k+1)>0}}$	sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)	Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]