11.9

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
11.9.E1	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{\mathrm{d}z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=z^{\mu-1}}}$ \deriv[2]{w}{z}+\frac{1}{z}\deriv{w}{z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w = z^{\mu-1}		diff(w, [z$(2)])+(1)/(z)*diff(w, z)+(1 -((nu)^(2))/((z)^(2)))*w = (z)^(mu - 1)	D[w, {z, 2}]+Divide[1,z]*D[w, z]+(1 -Divide[\[Nu]^(2),(z)^(2)])*w == (z)^(\[Mu]- 1)	Failure	Failure	Failed [300 / 300] Result: -.7677724761+.5394693872e-1*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 1.394855253+1.097179568*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.7677724760456721, 0.053946938885231305] 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[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.9642783315232053, 1.0539469388852312] 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[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
11.9.E2	${\displaystyle{\displaystyle w=s_{{\mu},{\nu}}\left(z\right)+AJ_{\nu}\left(z\right)+BY_{\nu}\left(z\right)}}$ w = \Lommels{\mu}{\nu}@{z}+A\BesselJ{\nu}@{z}+B\BesselY{\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	w = LommelS1(mu, nu, z)+ A*BesselJ(nu, z)+ B*BesselY(nu, z)	Error	Failure	Missing Macro Error	Failed [300 / 300] Result: .9752372341+.4710119144*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 1.486411080-.4774576789*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	-
11.9.E3	${\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=z^{\mu+1}\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{2k}}{a_{k+1}(\mu,\nu)}}}$ \Lommels{\mu}{\nu}@{z} = z^{\mu+1}\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{2k}}{a_{k+1}(\mu,\nu)}		LommelS1(mu, nu, z) = (z)^(mu + 1)* sum((- 1)^(k)*((z)^(2*k))/(a[k + 1](mu , nu)), k = 0..infinity)	Error	Failure	Missing Macro Error	Error	-
11.9.E4	${\displaystyle{\displaystyle a_{k}(\mu,\nu)=\prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)}}$ a_{k}(\mu,\nu) = \prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)		a[k](mu , nu) = product((mu + 2*m - 1)^(2)- (nu)^(2), m = 1..k)	Subscript[a, k][\[Mu], \[Nu]] == Product[(\[Mu]+ 2*m - 1)^(2)- \[Nu]^(2), {m, 1, k}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
11.9.E5	${\displaystyle{\displaystyle S_{{\mu},{\nu}}\left(z\right)=s_{{\mu},{\nu}}\left(z\right)+2^{\mu-1}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\*\left(\sin\left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,J_{\nu}\left(z\right)-\cos\left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,Y_{\nu}\left(z\right)\right)}}$ \LommelS{\mu}{\nu}@{z} = \Lommels{\mu}{\nu}@{z}+2^{\mu-1}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}\*\left(\sin@{\tfrac{1}{2}(\mu-\nu)\pi}\,\BesselJ{\nu}@{z}-\cos@{\tfrac{1}{2}(\mu-\nu)\pi}\,\BesselY{\nu}@{z}\right)	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})>0,\Re((-\nu)+k+1)>0}}$	LommelS2(mu, nu, z) = LommelS1(mu, nu, z)+ (2)^(mu - 1)* GAMMA((1)/(2)*mu +(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*mu -(1)/(2)*nu +(1)/(2))*(sin((1)/(2)*(mu - nu)*Pi)*BesselJ(nu, z)- cos((1)/(2)*(mu - nu)*Pi)*BesselY(nu, z))	Error	Successful	Missing Macro Error	-	-
11.9#Ex1	${\displaystyle{\displaystyle s_{{\mu},{-\nu}}\left(z\right)=s_{{\mu},{\nu}}\left(z\right)}}$ \Lommels{\mu}{-\nu}@{z} = \Lommels{\mu}{\nu}@{z}		LommelS1(mu, - nu, z) = LommelS1(mu, nu, z)	Error	Successful	Missing Macro Error	-	-
11.9#Ex2	${\displaystyle{\displaystyle S_{{\mu},{-\nu}}\left(z\right)=S_{{\mu},{\nu}}\left(z\right)}}$ \LommelS{\mu}{-\nu}@{z} = \LommelS{\mu}{\nu}@{z}	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}(-\nu)+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}(-\nu)+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})>0,\Re((-(-\nu))+k+1)>0}}$	LommelS2(mu, - nu, z) = LommelS2(mu, nu, z)	Error	Successful	Missing Macro Error	-	-
11.9.E7	${\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)\Gamma\left(k+\mu+1\right)}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}J_{2k+\mu+1}\left(z\right)}}$ \Lommels{\mu}{\nu}@{z} = 2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)\EulerGamma@{k+\mu+1}}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}\BesselJ{2k+\mu+1}@{z}	${\displaystyle{\displaystyle\Re((2k+\mu+1)+k+1)>0,\Re(k+\mu+1)>0}}$	LommelS1(mu, nu, z) = (2)^(mu + 1)* sum(*((2*k + mu + 1)*GAMMA(k + mu + 1))/(factorial(k)*(2*k + mu - nu + 1)*(2*k + mu + nu + 1))*BesselJ(2*k + mu + 1, z), k = 0..infinity)	Error	Error	Missing Macro Error	-	-
11.9.E8	${\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=2^{(\mu+\nu-1)/2}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)z^{(\mu+1-\nu)/2}\*\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}J_{k+\frac{1}{2}(\mu+\nu+1)}\left(z\right)}}$ \Lommels{\mu}{\nu}@{z} = 2^{(\mu+\nu-1)/2}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}z^{(\mu+1-\nu)/2}\*\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}\BesselJ{k+\frac{1}{2}(\mu+\nu+1)}@{z}	${\displaystyle{\displaystyle\Re((k+\frac{1}{2}(\mu+\nu+1))+k+1)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2})>0}}$	LommelS1(mu, nu, z) = (2)^((mu + nu - 1)/2)* GAMMA((1)/(2)*mu +(1)/(2)*nu +(1)/(2))*(z)^((mu + 1 - nu)/2)* sum((((1)/(2)*z)^(k))/(factorial(k)*(2*k + mu - nu + 1))*BesselJ(k +(1)/(2)*(mu + nu + 1), z), k = 0..infinity)	Error	Failure	Missing Macro Error	Skipped - Because timed out	-