10.54

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.54.E1	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin\theta)^{2n+1}\mathrm{d}\theta}}$ \sphBesselJ{n}@{z} = \frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2n+1}\diff{\theta}	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Divide[(z)^(n),(2)^(n + 1)* (n)!]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*n + 1), {\[Theta], 0, Pi}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.54.E2	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P_{n}\left(\cos\theta\right)\sin\theta\mathrm{d}\theta}}$ \sphBesselJ{n}@{z} = \frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\assLegendreP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta}	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Divide[(- I)^(n),2]*Integrate[Exp[I*z*Cos[\[Theta]]]*LegendreP[n, 0, 3, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Successful [Tested: 21]
10.54.E3	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=\frac{\pi}{2}\int_{1}^{\infty}e^{-zt}P_{n}\left(t\right)\mathrm{d}t}}$ \modsphBesselK{n}@{z} = \frac{\pi}{2}\int_{1}^{\infty}e^{-zt}\assLegendreP[]{n}@{t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi.}}$	Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Divide[Pi,2]*Integrate[Exp[- z*t]*LegendreP[n, 0, 3, t], {t, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
10.54.E4	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}Q_{n}\left(t\right)\mathrm{d}t}}$ \sphBesselJ{n}@{z} = \frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi.,\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Divide[(- I)^(n + 1),2*Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 + , 1 +)}, GenerateConditions->None]	Missing Macro Error	Failure	-	Error
10.54#Ex1	${\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}Q_{n}\left(t\right)\mathrm{d}t}}$ \sphHankelh{1}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}		Error	SphericalHankelH1[n, z] == Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (1 +)}, GenerateConditions->None]	Missing Macro Error	Failure	-	Error
10.54#Ex2	${\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}Q_{n}\left(t\right)\mathrm{d}t}}$ \sphHankelh{2}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi.}}$	Error	SphericalHankelH2[n, z] == Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 +)}, GenerateConditions->None]	Missing Macro Error	Failure	-	Error