Bessel Functions - 10.54 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
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Maple 		Numeric
Mathematica
10.54.E1 𝗃 n ⁑ ( z ) = z n 2 n + 1 ⁒ n ! ⁒ ∫ 0 Ο€ cos ⁑ ( z ⁒ cos ⁑ ΞΈ ) ⁒ ( sin ⁑ ΞΈ ) 2 ⁒ n + 1 ⁒ d ΞΈ spherical-Bessel-J 𝑛 𝑧 superscript 𝑧 𝑛 superscript 2 𝑛 1 𝑛 superscript subscript 0 πœ‹ 𝑧 πœƒ superscript πœƒ 2 𝑛 1 πœƒ {\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}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗃 n ⁑ ( z ) = ( - i ) n 2 ⁒ ∫ 0 Ο€ e i ⁒ z ⁒ cos ⁑ ΞΈ ⁒ P n ⁑ ( cos ⁑ ΞΈ ) ⁒ sin ⁑ ΞΈ ⁒ d ΞΈ spherical-Bessel-J 𝑛 𝑧 superscript 𝑖 𝑛 2 superscript subscript 0 πœ‹ superscript 𝑒 𝑖 𝑧 πœƒ shorthand-Legendre-P-first-kind 𝑛 πœƒ πœƒ πœƒ {\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}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗄 n ⁑ ( z ) = Ο€ 2 ⁒ ∫ 1 ∞ e - z ⁒ t ⁒ P n ⁑ ( t ) ⁒ d t spherical-Bessel-K 𝑛 𝑧 πœ‹ 2 superscript subscript 1 superscript 𝑒 𝑧 𝑑 shorthand-Legendre-P-first-kind 𝑛 𝑑 𝑑 {\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}		 | ph ⁑ z | < 1 2 ⁒ Ο€ . phase 𝑧 1 2 πœ‹ {\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 𝗃 n ⁑ ( z ) = ( - i ) n + 1 2 ⁒ Ο€ ⁒ ∫ i ⁒ ∞ ( - 1 + , 1 + ) e i ⁒ z ⁒ t ⁒ Q n ⁑ ( t ) ⁒ d t spherical-Bessel-J 𝑛 𝑧 superscript 𝑖 𝑛 1 2 πœ‹ superscript subscript 𝑖 limit-from 1 limit-from 1 superscript 𝑒 𝑖 𝑧 𝑑 shorthand-Legendre-Q-second-kind 𝑛 𝑑 𝑑 {\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}		 | ph ⁑ z | < 1 2 Ο€ . , β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 fragments | phase 𝑧 | 1 2 Ο€ . , 𝑛 1 2 π‘˜ 1 0 , 𝑛 1 2 π‘˜ 1 0 , 𝑛 1 2 π‘˜ 1 0 {\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 𝗁 n ( 1 ) ⁑ ( z ) = ( - i ) n + 1 Ο€ ⁒ ∫ i ⁒ ∞ ( 1 + ) e i ⁒ z ⁒ t ⁒ Q n ⁑ ( t ) ⁒ d t spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 superscript 𝑖 𝑛 1 πœ‹ superscript subscript 𝑖 limit-from 1 superscript 𝑒 𝑖 𝑧 𝑑 shorthand-Legendre-Q-second-kind 𝑛 𝑑 𝑑 {\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 𝗁 n ( 2 ) ⁑ ( z ) = ( - i ) n + 1 Ο€ ⁒ ∫ i ⁒ ∞ ( - 1 + ) e i ⁒ z ⁒ t ⁒ Q n ⁑ ( t ) ⁒ d t spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 superscript 𝑖 𝑛 1 πœ‹ superscript subscript 𝑖 limit-from 1 superscript 𝑒 𝑖 𝑧 𝑑 shorthand-Legendre-Q-second-kind 𝑛 𝑑 𝑑 {\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}		 | ph ⁑ z | < 1 2 ⁒ Ο€ . phase 𝑧 1 2 πœ‹ {\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
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