Bessel Functions - 10.9 Integral Representations

From testwiki
Revision as of 11:22, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.9.E1 J 0 ⁡ ( z ) = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ ) ⁢ d θ Bessel-J 0 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle J_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta\right)\mathrm{d}\theta}}
\BesselJ{0}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
BesselJ(0, z) = (1)/(Pi)*int(cos(z*sin(theta)), theta = 0..Pi)

BesselJ[0, z] == Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
Successful Successful -
Failed [4 / 7]
Result: Complex[0.1024204169391214, -0.20298051839359257]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.35155242920280916, 0.2300320660405755]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.9.E1 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ ) ⁢ d θ = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ cos ⁡ θ ) ⁢ d θ 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta% \right)\mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\cos\theta\right% )\mathrm{d}\theta}}
\frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0}} 
(1)/(Pi)*int(cos(z*sin(theta)), theta = 0..Pi) = (1)/(Pi)*int(cos(z*cos(theta)), theta = 0..Pi)

Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[1,Pi]*Integrate[Cos[z*Cos[\[Theta]]], {\[Theta], 0, Pi}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
10.9.E2 J n ⁡ ( z ) = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ - n ⁢ θ ) ⁢ d θ Bessel-J 𝑛 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝑛 𝜃 𝜃 {\displaystyle{\displaystyle J_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta-n\theta\right)\mathrm{d}\theta}}
\BesselJ{n}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
BesselJ(n, z) = (1)/(Pi)*int(cos(z*sin(theta)- n*theta), theta = 0..Pi)

BesselJ[n, z] == Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]
Failure Aborted Successful [Tested: 7] Successful [Tested: 7]
10.9.E2 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ - n ⁢ θ ) ⁢ d θ = i - n π ⁢ ∫ 0 π e i ⁢ z ⁢ cos ⁡ θ ⁢ cos ⁡ ( n ⁢ θ ) ⁢ d θ 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝑛 𝜃 𝜃 superscript 𝑖 𝑛 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝑖 𝑧 𝜃 𝑛 𝜃 𝜃 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-n% \theta\right)\mathrm{d}\theta=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos\theta}% \cos\left(n\theta\right)\mathrm{d}\theta}}
\frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta} = \frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\cos@{n\theta}\diff{\theta}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
(1)/(Pi)*int(cos(z*sin(theta)- n*theta), theta = 0..Pi) = ((I)^(- n))/(Pi)*int(exp(I*z*cos(theta))*cos(n*theta), theta = 0..Pi)

Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[(I)^(- n),Pi]*Integrate[Exp[I*z*Cos[\[Theta]]]*Cos[n*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]
Failure Aborted Successful [Tested: 7] Skipped - Because timed out
10.9.E3 Y 0 ⁡ ( z ) = 4 π 2 ⁢ ∫ 0 1 2 ⁢ π cos ⁡ ( z ⁢ cos ⁡ θ ) ⁢ ( γ + ln ⁡ ( 2 ⁢ z ⁢ sin 2 ⁡ θ ) ) ⁢ d θ Bessel-Y-Weber 0 𝑧 4 superscript 𝜋 2 superscript subscript 0 1 2 𝜋 𝑧 𝜃 2 𝑧 2 𝜃 𝜃 {\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{4}{\pi^{2}}\int_{0}^{% \frac{1}{2}\pi}\cos\left(z\cos\theta\right)\left(\gamma+\ln\left(2z{\sin^{2}}% \theta\right)\right)\mathrm{d}\theta}}
\BesselY{0}@{z} = \frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos@{z\cos@@{\theta}}\left(\EulerConstant+\ln@{2z\sin^{2}@@{\theta}}\right)\diff{\theta}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( - 0 ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0}} 
BesselY(0, z) = (4)/((Pi)^(2))*int(cos(z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..(1)/(2)*Pi)

BesselY[0, z] == Divide[4,(Pi)^(2)]*Integrate[Cos[z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Divide[1,2]*Pi}, GenerateConditions->None]
Aborted Aborted Successful [Tested: 7] Skipped - Because timed out
10.9.E4 J ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π cos ⁡ ( z ⁢ cos ⁡ θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left% (z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta}}
\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 1 2 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu+\tfrac% {1}{2})>0}} 
BesselJ(nu, z) = (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(cos(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)

BesselJ[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None]
Error Successful -
Failed [20 / 35]
Result: Complex[0.009683985979314524, -0.05759180507972181]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.21993206762171735, 0.08917811286212163]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
10.9.E4 ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 π cos ⁡ ( z ⁢ cos ⁡ θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ ν ⁢ d θ = 2 ⁢ ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 1 ( 1 - t 2 ) ν - 1 2 ⁢ cos ⁡ ( z ⁢ t ) ⁢ d t superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 𝜋 𝑧 𝜃 superscript 𝜃 2 𝜈 𝜃 2 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}% \Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left(z\cos\theta\right)% (\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1% }{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2% }}\cos\left(zt\right)\mathrm{d}t}}
\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2\nu}\diff{\theta} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\cos@{zt}\diff{t}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 1 2 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu+\tfrac% {1}{2})>0}} 
(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(cos(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi) = (2*((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* cos(z*t), t = 0..1)

Divide[(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}, GenerateConditions->None] == Divide[2*(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Cos[z*t], {t, 0, 1}, GenerateConditions->None]
Error Successful - Successful [Tested: 35]
10.9.E5 Y ν ⁡ ( z ) = 2 ⁢ ( 1 2 ⁢ z ) ν π 1 2 ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ( ∫ 0 1 ( 1 - t 2 ) ν - 1 2 ⁢ sin ⁡ ( z ⁢ t ) ⁢ d t - ∫ 0 ∞ e - z ⁢ t ⁢ ( 1 + t 2 ) ν - 1 2 ⁢ d t ) Bessel-Y-Weber 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑧 𝑡 𝑡 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑡 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu% }}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t% ^{2})^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t-\int_{0}^{\infty}e^{-zt}% (1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t\right)}}
\BesselY{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\nu+\tfrac{1}{2}}}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin@{zt}\diff{t}-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}\right)		 ℜ ⁡ ν > - 1 2 , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ν + 1 2 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 2 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1% }{2}\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(\nu+\tfrac{1}{2})>0}} 
BesselY(nu, z) = (2*((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*(int((1 - (t)^(2))^(nu -(1)/(2))* sin(z*t), t = 0..1)- int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity))

BesselY[\[Nu], z] == Divide[2*(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*(Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Sin[z*t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}, GenerateConditions->None])
Successful Successful -
Failed [15 / 25]
Result: Complex[-0.9495382353861556, 0.46093572348323536]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}


Result: Complex[-0.7706973036767981, 0.20650772012904162]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}

... skip entries to safe data
10.9.E6 J ν ⁡ ( z ) = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - sin ⁡ ( ν ⁢ π ) π ⁢ ∫ 0 ∞ e - z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t Bessel-J 𝜈 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜈 𝜃 𝜃 𝜈 𝜋 𝜋 superscript subscript 0 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi% \right)}{\pi}\int_{0}^{\infty}e^{-z\sinh t-\nu t}\mathrm{d}t}}
\BesselJ{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{\sin@{\nu\pi}}{\pi}\int_{0}^{\infty}e^{-z\sinh@@{t}-\nu t}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0}} 
BesselJ(nu, z) = (1)/(Pi)*int(cos(z*sin(theta)- nu*theta), theta = 0..Pi)-(sin(nu*Pi))/(Pi)*int(exp(- z*sinh(t)- nu*t), t = 0..infinity)

BesselJ[\[Nu], z] == Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[Sin[\[Nu]*Pi],Pi]*Integrate[Exp[- z*Sinh[t]- \[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [1 / 50]
Result: -.1812319652
Test Values: {nu = -1/2, z = 3/2}

Skipped - Because timed out
10.9.E7 Y ν ⁡ ( z ) = 1 π ⁢ ∫ 0 π sin ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - 1 π ⁢ ∫ 0 ∞ ( e ν ⁢ t + e - ν ⁢ t ⁢ cos ⁡ ( ν ⁢ π ) ) ⁢ e - z ⁢ sinh ⁡ t ⁢ d t Bessel-Y-Weber 𝜈 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜈 𝜃 𝜃 1 𝜋 superscript subscript 0 superscript 𝑒 𝜈 𝑡 superscript 𝑒 𝜈 𝑡 𝜈 𝜋 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{% \infty}\left(e^{\nu t}+e^{-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}% \mathrm{d}t}}
\BesselY{\nu}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\sin@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t}+e^{-\nu t}\cos@{\nu\pi}\right)e^{-z\sinh@@{t}}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0,\Re((-\nu)+k+1)>0}} 
BesselY(nu, z) = (1)/(Pi)*int(sin(z*sin(theta)- nu*theta), theta = 0..Pi)-(1)/(Pi)*int((exp(nu*t)+ exp(- nu*t)*cos(nu*Pi))*exp(- z*sinh(t)), t = 0..infinity)

BesselY[\[Nu], z] == Divide[1,Pi]*Integrate[Sin[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[1,Pi]*Integrate[(Exp[\[Nu]*t]+ Exp[- \[Nu]*t]*Cos[\[Nu]*Pi])*Exp[- z*Sinh[t]], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9#Ex1 J ν ⁡ ( x ) = 2 π ⁢ ∫ 0 ∞ sin ⁡ ( x ⁢ cosh ⁡ t - 1 2 ⁢ ν ⁢ π ) ⁢ cosh ⁡ ( ν ⁢ t ) ⁢ d t Bessel-J 𝜈 𝑥 2 𝜋 superscript subscript 0 𝑥 𝑡 1 2 𝜈 𝜋 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2}{\pi}\int_{0}^{% \infty}\sin\left(x\cosh t-\tfrac{1}{2}\nu\pi\right)\cosh\left(\nu t\right)% \mathrm{d}t}}
\BesselJ{\nu}@{x} = \frac{2}{\pi}\int_{0}^{\infty}\sin@{x\cosh@@{t}-\tfrac{1}{2}\nu\pi}\cosh@{\nu t}\diff{t}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, x) = (2)/(Pi)*int(sin(x*cosh(t)-(1)/(2)*nu*Pi)*cosh(nu*t), t = 0..infinity)

BesselJ[\[Nu], x] == Divide[2,Pi]*Integrate[Sin[x*Cosh[t]-Divide[1,2]*\[Nu]*Pi]*Cosh[\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9#Ex2 Y ν ⁡ ( x ) = - 2 π ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ cosh ⁡ t - 1 2 ⁢ ν ⁢ π ) ⁢ cosh ⁡ ( ν ⁢ t ) ⁢ d t Bessel-Y-Weber 𝜈 𝑥 2 𝜋 superscript subscript 0 𝑥 𝑡 1 2 𝜈 𝜋 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle Y_{\nu}\left(x\right)=-\frac{2}{\pi}\int_{0}^{% \infty}\cos\left(x\cosh t-\tfrac{1}{2}\nu\pi\right)\cosh\left(\nu t\right)% \mathrm{d}t}}
\BesselY{\nu}@{x} = -\frac{2}{\pi}\int_{0}^{\infty}\cos@{x\cosh@@{t}-\tfrac{1}{2}\nu\pi}\cosh@{\nu t}\diff{t}		 | ℜ ⁡ ν | < 1 , x > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence 𝑥 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\Re\nu|<1,x>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
BesselY(nu, x) = -(2)/(Pi)*int(cos(x*cosh(t)-(1)/(2)*nu*Pi)*cosh(nu*t), t = 0..infinity)

BesselY[\[Nu], x] == -Divide[2,Pi]*Integrate[Cos[x*Cosh[t]-Divide[1,2]*\[Nu]*Pi]*Cosh[\[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9#Ex3 J 0 ⁡ ( x ) = 2 π ⁢ ∫ 0 ∞ sin ⁡ ( x ⁢ cosh ⁡ t ) ⁢ d t Bessel-J 0 𝑥 2 𝜋 superscript subscript 0 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle J_{0}\left(x\right)=\frac{2}{\pi}\int_{0}^{\infty% }\sin\left(x\cosh t\right)\mathrm{d}t}}
\BesselJ{0}@{x} = \frac{2}{\pi}\int_{0}^{\infty}\sin@{x\cosh@@{t}}\diff{t}		 x > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 formulae-sequence 𝑥 0 0 𝑘 1 0 {\displaystyle{\displaystyle x>0,\Re(0+k+1)>0}} 
BesselJ(0, x) = (2)/(Pi)*int(sin(x*cosh(t)), t = 0..infinity)

BesselJ[0, x] == Divide[2,Pi]*Integrate[Sin[x*Cosh[t]], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9#Ex4 Y 0 ⁡ ( x ) = - 2 π ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ cosh ⁡ t ) ⁢ d t Bessel-Y-Weber 0 𝑥 2 𝜋 superscript subscript 0 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle Y_{0}\left(x\right)=-\frac{2}{\pi}\int_{0}^{% \infty}\cos\left(x\cosh t\right)\mathrm{d}t}}
\BesselY{0}@{x} = -\frac{2}{\pi}\int_{0}^{\infty}\cos@{x\cosh@@{t}}\diff{t}		 x > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( ( - 0 ) + k + 1 ) > 0 formulae-sequence 𝑥 0 formulae-sequence 0 𝑘 1 0 0 𝑘 1 0 {\displaystyle{\displaystyle x>0,\Re(0+k+1)>0,\Re((-0)+k+1)>0}} 
BesselY(0, x) = -(2)/(Pi)*int(cos(x*cosh(t)), t = 0..infinity)

BesselY[0, x] == -Divide[2,Pi]*Integrate[Cos[x*Cosh[t]], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9.E10 H ν ( 1 ) ⁡ ( z ) = e - 1 2 ⁢ ν ⁢ π ⁢ i π ⁢ i ⁢ ∫ - ∞ ∞ e i ⁢ z ⁢ cosh ⁡ t - ν ⁢ t ⁢ d t Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 1 2 𝜈 𝜋 𝑖 𝜋 𝑖 superscript subscript superscript 𝑒 𝑖 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{e^{-\frac{1}{2% }\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t-\nu t}\mathrm{d}t}}
\HankelH{1}{\nu}@{z} = \frac{e^{-\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{iz\cosh@@{t}-\nu t}\diff{t}		 0 < ph ⁡ z , ph ⁡ z < π formulae-sequence 0 phase 𝑧 phase 𝑧 𝜋 {\displaystyle{\displaystyle 0<\operatorname{ph}z,\operatorname{ph}z<\pi}} 
HankelH1(nu, z) = (exp(-(1)/(2)*nu*Pi*I))/(Pi*I)*int(exp(I*z*cosh(t)- nu*t), t = - infinity..infinity)

HankelH1[\[Nu], z] == Divide[Exp[-Divide[1,2]*\[Nu]*Pi*I],Pi*I]*Integrate[Exp[I*z*Cosh[t]- \[Nu]*t], {t, - Infinity, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9.E11 H ν ( 2 ) ⁡ ( z ) = - e 1 2 ⁢ ν ⁢ π ⁢ i π ⁢ i ⁢ ∫ - ∞ ∞ e - i ⁢ z ⁢ cosh ⁡ t - ν ⁢ t ⁢ d t Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 1 2 𝜈 𝜋 𝑖 𝜋 𝑖 superscript subscript superscript 𝑒 𝑖 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2% }\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-\nu t}\mathrm{d}t}}
\HankelH{2}{\nu}@{z} = -\frac{e^{\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh@@{t}-\nu t}\diff{t}		 - π < ph ⁡ z , ph ⁡ z < 0 formulae-sequence 𝜋 phase 𝑧 phase 𝑧 0 {\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<0}} 
HankelH2(nu, z) = -(exp((1)/(2)*nu*Pi*I))/(Pi*I)*int(exp(- I*z*cosh(t)- nu*t), t = - infinity..infinity)

HankelH2[\[Nu], z] == -Divide[Exp[Divide[1,2]*\[Nu]*Pi*I],Pi*I]*Integrate[Exp[- I*z*Cosh[t]- \[Nu]*t], {t, - Infinity, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9#Ex5 J ν ⁡ ( x ) = 2 ⁢ ( 1 2 ⁢ x ) - ν π 1 2 ⁢ Γ ⁡ ( 1 2 - ν ) ⁢ ∫ 1 ∞ sin ⁡ ( x ⁢ t ) ⁢ d t ( t 2 - 1 ) ν + 1 2 Bessel-J 𝜈 𝑥 2 superscript 1 2 𝑥 𝜈 superscript 𝜋 1 2 Euler-Gamma 1 2 𝜈 superscript subscript 1 𝑥 𝑡 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 {\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2(\tfrac{1}{2}x)^{-% \nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}% \frac{\sin\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}}}}
\BesselJ{\nu}@{x} = \frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\tfrac{1}{2}-\nu}}\int_{1}^{\infty}\frac{\sin@{xt}\diff{t}}{(t^{2}-1)^{\nu+\frac{1}{2}}}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 - ν ) > 0 formulae-sequence 𝜈 𝑘 1 0 1 2 𝜈 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}-\nu)>0}} 
BesselJ(nu, x) = (2*((1)/(2)*x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))*int((sin(x*t))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity)

BesselJ[\[Nu], x] == Divide[2*(Divide[1,2]*x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]*Integrate[Divide[Sin[x*t],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}, GenerateConditions->None]
Successful Aborted - Successful [Tested: 15]
10.9#Ex6 Y ν ⁡ ( x ) = - 2 ⁢ ( 1 2 ⁢ x ) - ν π 1 2 ⁢ Γ ⁡ ( 1 2 - ν ) ⁢ ∫ 1 ∞ cos ⁡ ( x ⁢ t ) ⁢ d t ( t 2 - 1 ) ν + 1 2 Bessel-Y-Weber 𝜈 𝑥 2 superscript 1 2 𝑥 𝜈 superscript 𝜋 1 2 Euler-Gamma 1 2 𝜈 superscript subscript 1 𝑥 𝑡 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 {\displaystyle{\displaystyle Y_{\nu}\left(x\right)=-\frac{2(\tfrac{1}{2}x)^{-% \nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}% \frac{\cos\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}}}}
\BesselY{\nu}@{x} = -\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\EulerGamma@{\tfrac{1}{2}-\nu}}\int_{1}^{\infty}\frac{\cos@{xt}\diff{t}}{(t^{2}-1)^{\nu+\frac{1}{2}}}		 | ℜ ⁡ ν | < 1 2 , x > 0 , ℜ ⁡ ( 1 2 - ν ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝑥 0 formulae-sequence 1 2 𝜈 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2},x>0,\Re(\tfrac{1}{2}-\nu)>0,% \Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
BesselY(nu, x) = -(2*((1)/(2)*x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))*int((cos(x*t))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity)

BesselY[\[Nu], x] == -Divide[2*(Divide[1,2]*x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]*Integrate[Divide[Cos[x*t],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}, GenerateConditions->None]
Successful Aborted - Skip - No test values generated
10.9.E13 ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ J ν ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = 1 π ⁢ ∫ 0 π e ζ ⁢ cos ⁡ θ ⁢ cos ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - sin ⁡ ( ν ⁢ π ) π ⁢ ∫ 0 ∞ e - ζ ⁢ cosh ⁡ t - z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Bessel-J 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝜁 𝜃 𝑧 𝜃 𝜈 𝜃 𝜃 𝜈 𝜋 𝜋 superscript subscript 0 superscript 𝑒 𝜁 𝑡 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}J_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t% -\nu t}\mathrm{d}t}}
\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\BesselJ{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = \frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos@@{\theta}}\cos@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{\sin@{\nu\pi}}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh@@{t}-z\sinh@@{t}-\nu t}\diff{t}		 ℜ ⁡ ( z + ζ ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝑧 𝜁 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(z+\zeta\right)>0,\Re(\nu+k+1)>0}} 
((z + zeta)/(z - zeta))^((1)/(2)*nu)* BesselJ(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = (1)/(Pi)*int(exp(zeta*cos(theta))*cos(z*sin(theta)- nu*theta), theta = 0..Pi)-(sin(nu*Pi))/(Pi)*int(exp(- zeta*cosh(t)- z*sinh(t)- nu*t), t = 0..infinity)

(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]*\[Nu])* BesselJ[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == Divide[1,Pi]*Integrate[Exp[\[Zeta]*Cos[\[Theta]]]*Cos[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[Sin[\[Nu]*Pi],Pi]*Integrate[Exp[- \[Zeta]*Cosh[t]- z*Sinh[t]- \[Nu]*t], {t, 0, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9.E14 ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ Y ν ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = 1 π ⁢ ∫ 0 π e ζ ⁢ cos ⁡ θ ⁢ sin ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - 1 π ⁢ ∫ 0 ∞ ( e ν ⁢ t + ζ ⁢ cosh ⁡ t + e - ν ⁢ t - ζ ⁢ cosh ⁡ t ⁢ cos ⁡ ( ν ⁢ π ) ) ⁢ e - z ⁢ sinh ⁡ t ⁢ d t superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Bessel-Y-Weber 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝜁 𝜃 𝑧 𝜃 𝜈 𝜃 𝜃 1 𝜋 superscript subscript 0 superscript 𝑒 𝜈 𝑡 𝜁 𝑡 superscript 𝑒 𝜈 𝑡 𝜁 𝑡 𝜈 𝜋 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}Y_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t+\zeta\cosh t}+e^{-\nu t-\zeta% \cosh t}\cos\left(\nu\pi\right)\right)\*e^{-z\sinh t}\mathrm{d}t}}
\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\BesselY{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = \frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos@@{\theta}}\sin@{z\sin@@{\theta}-\nu\theta}\diff{\theta}-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t+\zeta\cosh@@{t}}+e^{-\nu t-\zeta\cosh@@{t}}\cos@{\nu\pi}\right)\*e^{-z\sinh@@{t}}\diff{t}		 ℜ ⁡ ( z + ζ ) > 0 , ℜ ⁡ ( z - ζ ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝑧 𝜁 0 formulae-sequence 𝑧 𝜁 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(z+\zeta\right)>0,\Re\left(z-\zeta\right)>% 0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
((z + zeta)/(z - zeta))^((1)/(2)*nu)* BesselY(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = (1)/(Pi)*int(exp(zeta*cos(theta))*sin(z*sin(theta)- nu*theta), theta = 0..Pi)-(1)/(Pi)*int((exp(nu*t + zeta*cosh(t))+ exp(- nu*t - zeta*cosh(t))*cos(nu*Pi))* exp(- z*sinh(t)), t = 0..infinity)

(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]*\[Nu])* BesselY[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == Divide[1,Pi]*Integrate[Exp[\[Zeta]*Cos[\[Theta]]]*Sin[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]-Divide[1,Pi]*Integrate[(Exp[\[Nu]*t + \[Zeta]*Cosh[t]]+ Exp[- \[Nu]*t - \[Zeta]*Cosh[t]]*Cos[\[Nu]*Pi])* Exp[- z*Sinh[t]], {t, 0, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9.E15 ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ H ν ( 1 ) ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = 1 π ⁢ i ⁢ e - 1 2 ⁢ ν ⁢ π ⁢ i ⁢ ∫ - ∞ ∞ e i ⁢ z ⁢ cosh ⁡ t + i ⁢ ζ ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Hankel-H-1-Bessel-third-kind 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 𝑖 superscript 𝑒 1 2 𝜈 𝜋 𝑖 superscript subscript superscript 𝑒 𝑖 𝑧 𝑡 𝑖 𝜁 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}{H^{(1)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi i% }e^{-\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t+i\zeta\sinh t-\nu t% }\mathrm{d}t}}
\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\HankelH{1}{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = \frac{1}{\pi i}e^{-\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{iz\cosh@@{t}+i\zeta\sinh@@{t}-\nu t}\diff{t}		 

((z + zeta)/(z - zeta))^((1)/(2)*nu)* HankelH1(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = (1)/(Pi*I)*exp(-(1)/(2)*nu*Pi*I)*int(exp(I*z*cosh(t)+ I*zeta*sinh(t)- nu*t), t = - infinity..infinity)

(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]*\[Nu])* HankelH1[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == Divide[1,Pi*I]*Exp[-Divide[1,2]*\[Nu]*Pi*I]*Integrate[Exp[I*z*Cosh[t]+ I*\[Zeta]*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9.E16 ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ H ν ( 2 ) ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = - 1 π ⁢ i ⁢ e 1 2 ⁢ ν ⁢ π ⁢ i ⁢ ∫ - ∞ ∞ e - i ⁢ z ⁢ cosh ⁡ t - i ⁢ ζ ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Hankel-H-2-Bessel-third-kind 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 𝑖 superscript 𝑒 1 2 𝜈 𝜋 𝑖 superscript subscript superscript 𝑒 𝑖 𝑧 𝑡 𝑖 𝜁 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}{H^{(2)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i% }e^{\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t% }\mathrm{d}t}}
\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}\HankelH{2}{\nu}@{(z^{2}-\zeta^{2})^{\frac{1}{2}}} = -\frac{1}{\pi i}e^{\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh@@{t}-i\zeta\sinh@@{t}-\nu t}\diff{t}		 ℑ ⁡ ( z + ζ ) < 0 , ℑ ⁡ ( z - ζ ) < 0 formulae-sequence 𝑧 𝜁 0 𝑧 𝜁 0 {\displaystyle{\displaystyle\Im\left(z+\zeta\right)<0,\Im\left(z-\zeta\right)<% 0}} 
((z + zeta)/(z - zeta))^((1)/(2)*nu)* HankelH2(nu, ((z)^(2)- (zeta)^(2))^((1)/(2))) = -(1)/(Pi*I)*exp((1)/(2)*nu*Pi*I)*int(exp(- I*z*cosh(t)- I*zeta*sinh(t)- nu*t), t = - infinity..infinity)

(Divide[z + \[Zeta],z - \[Zeta]])^(Divide[1,2]*\[Nu])* HankelH2[\[Nu], ((z)^(2)- \[Zeta]^(2))^(Divide[1,2])] == -Divide[1,Pi*I]*Exp[Divide[1,2]*\[Nu]*Pi*I]*Integrate[Exp[- I*z*Cosh[t]- I*\[Zeta]*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9.E17 J ν ⁡ ( z ) = 1 2 ⁢ π ⁢ i ⁢ ∫ ∞ - π ⁢ i ∞ + π ⁢ i e z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t Bessel-J 𝜈 𝑧 1 2 𝜋 𝑖 superscript subscript 𝜋 𝑖 𝜋 𝑖 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{% \infty-\pi i}^{\infty+\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}
\BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, z) = (1)/(2*Pi*I)*int(exp(z*sinh(t)- nu*t), t = infinity - Pi*I..infinity + Pi*I)

BesselJ[\[Nu], z] == Divide[1,2*Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, Infinity - Pi*I, Infinity + Pi*I}, GenerateConditions->None]
Error Failure -
Failed [70 / 70]
Result: Complex[0.4358908643715884, -0.07192294931339177]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0679098760861825, 0.09257666026367889]
Test Values: {Rule[z, 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
10.9#Ex7 H ν ( 1 ) ⁡ ( z ) = 1 π ⁢ i ⁢ ∫ - ∞ ∞ + π ⁢ i e z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t Hankel-H-1-Bessel-third-kind 𝜈 𝑧 1 𝜋 𝑖 superscript subscript 𝜋 𝑖 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{1}{\pi i}\int_% {-\infty}^{\infty+\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}
\HankelH{1}{\nu}@{z} = \frac{1}{\pi i}\int_{-\infty}^{\infty+\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}		 

HankelH1(nu, z) = (1)/(Pi*I)*int(exp(z*sinh(t)- nu*t), t = - infinity..infinity + Pi*I)

HankelH1[\[Nu], z] == Divide[1,Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity + Pi*I}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9#Ex8 H ν ( 2 ) ⁡ ( z ) = - 1 π ⁢ i ⁢ ∫ - ∞ ∞ - π ⁢ i e z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t Hankel-H-2-Bessel-third-kind 𝜈 𝑧 1 𝜋 𝑖 superscript subscript 𝜋 𝑖 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{1}{\pi i}\int% _{-\infty}^{\infty-\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}
\HankelH{2}{\nu}@{z} = -\frac{1}{\pi i}\int_{-\infty}^{\infty-\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}		 

HankelH2(nu, z) = -(1)/(Pi*I)*int(exp(z*sinh(t)- nu*t), t = - infinity..infinity - Pi*I)

HankelH2[\[Nu], z] == -Divide[1,Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity - Pi*I}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
10.9.E19 J ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) exp ⁡ ( t - z 2 4 ⁢ t ) ⁢ d t t ν + 1 Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 2 𝜋 𝑖 superscript subscript limit-from 0 𝑡 superscript 𝑧 2 4 𝑡 𝑡 superscript 𝑡 𝜈 1 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{2\pi i}\int_{-\infty}^{(0+)}\exp\left(t-\frac{z^{2}}{4t}\right)\frac{\mathrm% {d}t}{t^{\nu+1}}}}
\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{2\pi i}\int_{-\infty}^{(0+)}\exp@{t-\frac{z^{2}}{4t}}\frac{\diff{t}}{t^{\nu+1}}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
BesselJ(nu, z) = (((1)/(2)*z)^(nu))/(2*Pi*I)*int(exp(t -((z)^(2))/(4*t))*(1)/((t)^(nu + 1)), t = - infinity..(0 +))

BesselJ[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],2*Pi*I]*Integrate[Exp[t -Divide[(z)^(2),4*t]]*Divide[1,(t)^(\[Nu]+ 1)], {t, - Infinity, (0 +)}, GenerateConditions->None]
Error Failure - Error
10.9.E20 J ν ⁡ ( z ) = Γ ⁡ ( 1 2 - ν ) ⁢ ( 1 2 ⁢ z ) ν π 3 2 ⁢ i ⁢ ∫ 0 ( 1 + ) cos ⁡ ( z ⁢ t ) ⁢ ( t 2 - 1 ) ν - 1 2 ⁢ d t Bessel-J 𝜈 𝑧 Euler-Gamma 1 2 𝜈 superscript 1 2 𝑧 𝜈 superscript 𝜋 3 2 𝑖 superscript subscript 0 limit-from 1 𝑧 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{% 2}-\nu\right)(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left% (zt\right)(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}
\BesselJ{\nu}@{z} = \frac{\EulerGamma@{\frac{1}{2}-\nu}(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos@{zt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}		 ν ≠ 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 - ν ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 1 2 𝜈 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\frac{1}{2}% -\nu)>0}} 
BesselJ(nu, z) = (GAMMA((1)/(2)- nu)*((1)/(2)*z)^(nu))/((Pi)^((3)/(2))* I)*int(cos(z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 0..(1 +))

BesselJ[\[Nu], z] == Divide[Gamma[Divide[1,2]- \[Nu]]*(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[3,2])* I]*Integrate[Cos[z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 0, (1 +)}, GenerateConditions->None]
Error Failure - Error
10.9#Ex9 H ν ( 1 ) ⁡ ( z ) = Γ ⁡ ( 1 2 - ν ) ⁢ ( 1 2 ⁢ z ) ν π 3 2 ⁢ i ⁢ ∫ 1 + i ⁢ ∞ ( 1 + ) e i ⁢ z ⁢ t ⁢ ( t 2 - 1 ) ν - 1 2 ⁢ d t Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Euler-Gamma 1 2 𝜈 superscript 1 2 𝑧 𝜈 superscript 𝜋 3 2 𝑖 superscript subscript 1 𝑖 limit-from 1 superscript 𝑒 𝑖 𝑧 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 𝑡 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{\Gamma\left(% \tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1+i% \infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}
\HankelH{1}{\nu}@{z} = \frac{\EulerGamma@{\tfrac{1}{2}-\nu}(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1+i\infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}		 ν ≠ 1 2 , 3 2 < 1 2 ⁢ π , … < 1 2 ⁢ π , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( 1 2 - ν ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 3 2 1 2 𝜋 formulae-sequence … 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 1 2 𝜈 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2}<\tfrac{1}{2}\pi,% \ldots<\tfrac{1}{2}\pi,|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\tfrac{1}{2}-% \nu)>0}} 
HankelH1(nu, z) = (GAMMA((1)/(2)- nu)*((1)/(2)*z)^(nu))/((Pi)^((3)/(2))* I)*int(exp(I*z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 1 + I*infinity..(1 +))

HankelH1[\[Nu], z] == Divide[Gamma[Divide[1,2]- \[Nu]]*(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[3,2])* I]*Integrate[Exp[I*z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1 + I*Infinity, (1 +)}, GenerateConditions->None]
Error Failure - Error
10.9#Ex10 H ν ( 2 ) ⁡ ( z ) = Γ ⁡ ( 1 2 - ν ) ⁢ ( 1 2 ⁢ z ) ν π 3 2 ⁢ i ⁢ ∫ 1 - i ⁢ ∞ ( 1 + ) e - i ⁢ z ⁢ t ⁢ ( t 2 - 1 ) ν - 1 2 ⁢ d t Hankel-H-2-Bessel-third-kind 𝜈 𝑧 Euler-Gamma 1 2 𝜈 superscript 1 2 𝑧 𝜈 superscript 𝜋 3 2 𝑖 superscript subscript 1 𝑖 limit-from 1 superscript 𝑒 𝑖 𝑧 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 𝑡 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=\frac{\Gamma\left(% \tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1-i% \infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}
\HankelH{2}{\nu}@{z} = \frac{\EulerGamma@{\tfrac{1}{2}-\nu}(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1-i\infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}\diff{t}		 ν ≠ 1 2 , 3 2 < 1 2 ⁢ π , … < 1 2 ⁢ π , | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( 1 2 - ν ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 3 2 1 2 𝜋 formulae-sequence … 1 2 𝜋 formulae-sequence phase 𝑧 1 2 𝜋 1 2 𝜈 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2}<\tfrac{1}{2}\pi,% \ldots<\tfrac{1}{2}\pi,|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\tfrac{1}{2}-% \nu)>0}} 
HankelH2(nu, z) = (GAMMA((1)/(2)- nu)*((1)/(2)*z)^(nu))/((Pi)^((3)/(2))* I)*int(exp(- I*z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 1 - I*infinity..(1 +))

HankelH2[\[Nu], z] == Divide[Gamma[Divide[1,2]- \[Nu]]*(Divide[1,2]*z)^\[Nu],(Pi)^(Divide[3,2])* I]*Integrate[Exp[- I*z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1 - I*Infinity, (1 +)}, GenerateConditions->None]
Error Failure - Error
10.9.E22 J ν ⁡ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( - t ) ⁢ ( 1 2 ⁢ x ) ν + 2 ⁢ t Γ ⁡ ( ν + t + 1 ) ⁢ d t Bessel-J 𝜈 𝑥 1 2 𝜋 𝑖 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 superscript 1 2 𝑥 𝜈 2 𝑡 Euler-Gamma 𝜈 𝑡 1 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i% \infty}^{i\infty}\frac{\Gamma\left(-t\right)(\tfrac{1}{2}x)^{\nu+2t}}{\Gamma% \left(\nu+t+1\right)}\mathrm{d}t}}
\BesselJ{\nu}@{x} = \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\EulerGamma@{-t}(\tfrac{1}{2}x)^{\nu+2t}}{\EulerGamma@{\nu+t+1}}\diff{t}		 ℜ ⁡ ν > 0 , x > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( - t ) > 0 , ℜ ⁡ ( ν + t + 1 ) > 0 formulae-sequence 𝜈 0 formulae-sequence 𝑥 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝑡 0 𝜈 𝑡 1 0 {\displaystyle{\displaystyle\Re\nu>0,x>0,\Re(\nu+k+1)>0,\Re(-t)>0,\Re(\nu+t+1)% >0}} 
BesselJ(nu, x) = (1)/(2*Pi*I)*int((GAMMA(- t)*((1)/(2)*x)^(nu + 2*t))/(GAMMA(nu + t + 1)), t = - I*infinity..I*infinity)

BesselJ[\[Nu], x] == Divide[1,2*Pi*I]*Integrate[Divide[Gamma[- t]*(Divide[1,2]*x)^(\[Nu]+ 2*t),Gamma[\[Nu]+ t + 1]], {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.9.E23 J ν ⁡ ( z ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ - i ⁢ c - ∞ + i ⁢ c Γ ⁡ ( t ) Γ ⁡ ( ν - t + 1 ) ⁢ ( 1 2 ⁢ z ) ν - 2 ⁢ t ⁢ d t Bessel-J 𝜈 𝑧 1 2 𝜋 𝑖 superscript subscript 𝑖 𝑐 𝑖 𝑐 Euler-Gamma 𝑡 Euler-Gamma 𝜈 𝑡 1 superscript 1 2 𝑧 𝜈 2 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{-% \infty-ic}^{-\infty+ic}\frac{\Gamma\left(t\right)}{\Gamma\left(\nu-t+1\right)}% (\tfrac{1}{2}z)^{\nu-2t}\mathrm{d}t}}
\BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{-\infty-ic}^{-\infty+ic}\frac{\EulerGamma@{t}}{\EulerGamma@{\nu-t+1}}(\tfrac{1}{2}z)^{\nu-2t}\diff{t}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ t > 0 , ℜ ⁡ ( ν - t + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝑡 0 𝜈 𝑡 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re t>0,\Re(\nu-t+1)>0}} 
BesselJ(nu, z) = (1)/(2*Pi*I)*int((GAMMA(t))/(GAMMA(nu - t + 1))*((1)/(2)*z)^(nu - 2*t), t = - infinity - I*c..- infinity + I*c)
 
BesselJ[\[Nu], z] == Divide[1,2*Pi*I]*Integrate[Divide[Gamma[t],Gamma[\[Nu]- t + 1]]*(Divide[1,2]*z)^(\[Nu]- 2*t), {t, - Infinity - I*c, - Infinity + I*c}, GenerateConditions->None]
 Failure Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[0.4358908643715884, -0.07192294931339177]
Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.0679098760861825, 0.09257666026367889]
Test Values: {Rule[c, -1.5], Rule[z, 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
10.9.E24 H ν ( 1 ) ⁡ ( z ) = - e - 1 2 ⁢ ν ⁢ π ⁢ i 2 ⁢ π 2 ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( t - ν ) ⁢ ( - 1 2 ⁢ i ⁢ z ) ν - 2 ⁢ t ⁢ d t Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 1 2 𝜈 𝜋 𝑖 2 superscript 𝜋 2 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 Euler-Gamma 𝑡 𝜈 superscript 1 2 𝑖 𝑧 𝜈 2 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=-\frac{e^{-\frac{1}{% 2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma% \left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t}}
\HankelH{1}{\nu}@{z} = -\frac{e^{-\frac{1}{2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+i\infty}\EulerGamma@{t}\EulerGamma@{t-\nu}(-\tfrac{1}{2}iz)^{\nu-2t}\diff{t}		 0 < ph ⁡ z , ph ⁡ z < π , ℜ ⁡ t > 0 , ℜ ⁡ ( t - ν ) > 0 formulae-sequence 0 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑡 0 𝑡 𝜈 0 {\displaystyle{\displaystyle 0<\operatorname{ph}z,\operatorname{ph}z<\pi,\Re t% >0,\Re(t-\nu)>0}} 
HankelH1(nu, z) = -(exp(-(1)/(2)*nu*Pi*I))/(2*(Pi)^(2))* int(GAMMA(t)*GAMMA(t - nu)*(-(1)/(2)*I*z)^(nu - 2*t), t = c - I*infinity..c + I*infinity)
 
HankelH1[\[Nu], z] == -Divide[Exp[-Divide[1,2]*\[Nu]*Pi*I],2*(Pi)^(2)]* Integrate[Gamma[t]*Gamma[t - \[Nu]]*(-Divide[1,2]*I*z)^(\[Nu]- 2*t), {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Failure Aborted
Failed [120 / 120]
Result: .2971181619-.8401954886*I
Test Values: {c = -3/2, nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -.8661908042+.2691615148*I
Test Values: {c = -3/2, nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
10.9.E25 H ν ( 2 ) ⁡ ( z ) = e 1 2 ⁢ ν ⁢ π ⁢ i 2 ⁢ π 2 ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( t - ν ) ⁢ ( 1 2 ⁢ i ⁢ z ) ν - 2 ⁢ t ⁢ d t Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 1 2 𝜈 𝜋 𝑖 2 superscript 𝜋 2 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 Euler-Gamma 𝑡 𝜈 superscript 1 2 𝑖 𝑧 𝜈 2 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=\frac{e^{\frac{1}{2}% \nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma% \left(t-\nu\right)(\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t}}
\HankelH{2}{\nu}@{z} = \frac{e^{\frac{1}{2}\nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{t}\EulerGamma@{t-\nu}(\tfrac{1}{2}iz)^{\nu-2t}\diff{t}		 - π < ph ⁡ z , ph ⁡ z < 0 , ℜ ⁡ t > 0 , ℜ ⁡ ( t - ν ) > 0 formulae-sequence 𝜋 phase 𝑧 formulae-sequence phase 𝑧 0 formulae-sequence 𝑡 0 𝑡 𝜈 0 {\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<0,\Re t% >0,\Re(t-\nu)>0}} 
HankelH2(nu, z) = (exp((1)/(2)*nu*Pi*I))/(2*(Pi)^(2))*int(GAMMA(t)*GAMMA(t - nu)*((1)/(2)*I*z)^(nu - 2*t), t = c - I*infinity..c + I*infinity)
 
HankelH2[\[Nu], z] == Divide[Exp[Divide[1,2]*\[Nu]*Pi*I],2*(Pi)^(2)]*Integrate[Gamma[t]*Gamma[t - \[Nu]]*(Divide[1,2]*I*z)^(\[Nu]- 2*t), {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Failure Aborted
Failed [120 / 120]
Result: -.1414870617+.1246394392*I
Test Values: {c = -3/2, nu = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

Result: -.1498748781e-1-.1846515642*I
Test Values: {c = -3/2, nu = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
 Skipped - Because timed out
10.9.E26 J μ ⁡ ( z ) ⁢ J ν ⁡ ( z ) = 2 π ⁢ ∫ 0 π / 2 J μ + ν ⁡ ( 2 ⁢ z ⁢ cos ⁡ θ ) ⁢ cos ⁡ ( ( μ - ν ) ⁢ θ ) ⁢ d θ Bessel-J 𝜇 𝑧 Bessel-J 𝜈 𝑧 2 𝜋 superscript subscript 0 𝜋 2 Bessel-J 𝜇 𝜈 2 𝑧 𝜃 𝜇 𝜈 𝜃 𝜃 {\displaystyle{\displaystyle J_{\mu}\left(z\right)J_{\nu}\left(z\right)=\frac{% 2}{\pi}\int_{0}^{\pi/2}J_{\mu+\nu}\left(2z\cos\theta\right)\cos\left((\mu-\nu)% \theta\right)\mathrm{d}\theta}}
\BesselJ{\mu}@{z}\BesselJ{\nu}@{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\BesselJ{\mu+\nu}@{2z\cos@@{\theta}}\cos@{(\mu-\nu)\theta}\diff{\theta}		 ℜ ⁡ ( μ + ν ) > - 1 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( μ + ν ) + k + 1 ) > 0 formulae-sequence 𝜇 𝜈 1 formulae-sequence 𝜇 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 𝜇 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>-1,\Re((\mu)+k+1)>0,\Re(% \nu+k+1)>0,\Re((\mu+\nu)+k+1)>0}} 
BesselJ(mu, z)*BesselJ(nu, z) = (2)/(Pi)*int(BesselJ(mu + nu, 2*z*cos(theta))*cos((mu - nu)*theta), theta = 0..Pi/2)
 
BesselJ[\[Mu], z]*BesselJ[\[Nu], z] == Divide[2,Pi]*Integrate[BesselJ[\[Mu]+ \[Nu], 2*z*Cos[\[Theta]]]*Cos[(\[Mu]- \[Nu])*\[Theta]], {\[Theta], 0, Pi/2}, GenerateConditions->None]
 Failure Aborted Manual Skip! Skipped - Because timed out
10.9.E27 J ν ⁡ ( z ) ⁢ J ν ⁡ ( ζ ) = 2 π ⁢ ∫ 0 π / 2 J 2 ⁢ ν ⁡ ( 2 ⁢ ( z ⁢ ζ ) 1 2 ⁢ sin ⁡ θ ) ⁢ cos ⁡ ( ( z - ζ ) ⁢ cos ⁡ θ ) ⁢ d θ Bessel-J 𝜈 𝑧 Bessel-J 𝜈 𝜁 2 𝜋 superscript subscript 0 𝜋 2 Bessel-J 2 𝜈 2 superscript 𝑧 𝜁 1 2 𝜃 𝑧 𝜁 𝜃 𝜃 {\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=% \frac{2}{\pi}\int_{0}^{\pi/2}J_{2\nu}\left(2(z\zeta)^{\frac{1}{2}}\sin\theta% \right)\cos\left((z-\zeta)\cos\theta\right)\mathrm{d}\theta}}
\BesselJ{\nu}@{z}\BesselJ{\nu}@{\zeta} = \frac{2}{\pi}\int_{0}^{\pi/2}\BesselJ{2\nu}@{2(z\zeta)^{\frac{1}{2}}\sin@@{\theta}}\cos@{(z-\zeta)\cos@@{\theta}}\diff{\theta}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ ν ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 2 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((2\nu)+k+1% )>0}} 
BesselJ(nu, z)*BesselJ(nu, zeta) = (2)/(Pi)*int(BesselJ(2*nu, 2*(z*zeta)^((1)/(2))* sin(theta))*cos((z - zeta)*cos(theta)), theta = 0..Pi/2)
 
BesselJ[\[Nu], z]*BesselJ[\[Nu], \[Zeta]] == Divide[2,Pi]*Integrate[BesselJ[2*\[Nu], 2*(z*\[Zeta])^(Divide[1,2])* Sin[\[Theta]]]*Cos[(z - \[Zeta])*Cos[\[Theta]]], {\[Theta], 0, Pi/2}, GenerateConditions->None]
 Failure Aborted Manual Skip! Skipped - Because timed out
10.9.E28 J ν ⁡ ( z ) ⁢ J ν ⁡ ( ζ ) = 1 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ ⁢ exp ⁡ ( 1 2 ⁢ t - z 2 + ζ 2 2 ⁢ t ) ⁢ I ν ⁡ ( z ⁢ ζ t ) ⁢ d t t Bessel-J 𝜈 𝑧 Bessel-J 𝜈 𝜁 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 1 2 𝑡 superscript 𝑧 2 superscript 𝜁 2 2 𝑡 modified-Bessel-first-kind 𝜈 𝑧 𝜁 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=% \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\*\exp\left(\frac{1}{2}t-\frac{z^{% 2}+\zeta^{2}}{2t}\right)I_{\nu}\left(\frac{z\zeta}{t}\right)\frac{\mathrm{d}t}% {t}}}
\BesselJ{\nu}@{z}\BesselJ{\nu}@{\zeta} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\*\exp@{\frac{1}{2}t-\frac{z^{2}+\zeta^{2}}{2t}}\modBesselI{\nu}@{\frac{z\zeta}{t}}\frac{\diff{t}}{t}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re(\nu+k+1)>0}} 
BesselJ(nu, z)*BesselJ(nu, zeta) = (1)/(2*Pi*I)*int(* exp((1)/(2)*t -((z)^(2)+ (zeta)^(2))/(2*t))*BesselI(nu, (z*zeta)/(t))*(1)/(t), t = c - I*infinity..c + I*infinity)
 
BesselJ[\[Nu], z]*BesselJ[\[Nu], \[Zeta]] == Divide[1,2*Pi*I]*Integrate[* Exp[Divide[1,2]*t -Divide[(z)^(2)+ \[Zeta]^(2),2*t]]*BesselI[\[Nu], Divide[z*\[Zeta],t]]*Divide[1,t], {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]
 Error Failure - Error
10.9.E29 J μ ⁡ ( x ) ⁢ J ν ⁡ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( - t ) ⁢ Γ ⁡ ( 2 ⁢ t + μ + ν + 1 ) ⁢ ( 1 2 ⁢ x ) μ + ν + 2 ⁢ t Γ ⁡ ( t + μ + 1 ) ⁢ Γ ⁡ ( t + ν + 1 ) ⁢ Γ ⁡ ( t + μ + ν + 1 ) ⁢ d t Bessel-J 𝜇 𝑥 Bessel-J 𝜈 𝑥 1 2 𝜋 𝑖 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 2 𝑡 𝜇 𝜈 1 superscript 1 2 𝑥 𝜇 𝜈 2 𝑡 Euler-Gamma 𝑡 𝜇 1 Euler-Gamma 𝑡 𝜈 1 Euler-Gamma 𝑡 𝜇 𝜈 1 𝑡 {\displaystyle{\displaystyle J_{\mu}\left(x\right)J_{\nu}\left(x\right)=\frac{% 1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+% \mu+\nu+1\right)(\tfrac{1}{2}x)^{\mu+\nu+2t}}{\Gamma\left(t+\mu+1\right)\Gamma% \left(t+\nu+1\right)\Gamma\left(t+\mu+\nu+1\right)}\mathrm{d}t}}
\BesselJ{\mu}@{x}\BesselJ{\nu}@{x} = \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\EulerGamma@{-t}\EulerGamma@{2t+\mu+\nu+1}(\tfrac{1}{2}x)^{\mu+\nu+2t}}{\EulerGamma@{t+\mu+1}\EulerGamma@{t+\nu+1}\EulerGamma@{t+\mu+\nu+1}}\diff{t}		 x > 0 , ℜ ⁡ ( ( μ ) + k + 1 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( - t ) > 0 , ℜ ⁡ ( 2 ⁢ t + μ + ν + 1 ) > 0 , ℜ ⁡ ( t + μ + 1 ) > 0 , ℜ ⁡ ( t + ν + 1 ) > 0 , ℜ ⁡ ( t + μ + ν + 1 ) > 0 formulae-sequence 𝑥 0 formulae-sequence 𝜇 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝑡 0 formulae-sequence 2 𝑡 𝜇 𝜈 1 0 formulae-sequence 𝑡 𝜇 1 0 formulae-sequence 𝑡 𝜈 1 0 𝑡 𝜇 𝜈 1 0 {\displaystyle{\displaystyle x>0,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,\Re(-t)>0,\Re% (2t+\mu+\nu+1)>0,\Re(t+\mu+1)>0,\Re(t+\nu+1)>0,\Re(t+\mu+\nu+1)>0}} 
BesselJ(mu, x)*BesselJ(nu, x) = (1)/(2*Pi*I)*int((GAMMA(- t)*GAMMA(2*t + mu + nu + 1)*((1)/(2)*x)^(mu + nu + 2*t))/(GAMMA(t + mu + 1)*GAMMA(t + nu + 1)*GAMMA(t + mu + nu + 1)), t = - I*infinity..I*infinity)
 
BesselJ[\[Mu], x]*BesselJ[\[Nu], x] == Divide[1,2*Pi*I]*Integrate[Divide[Gamma[- t]*Gamma[2*t + \[Mu]+ \[Nu]+ 1]*(Divide[1,2]*x)^(\[Mu]+ \[Nu]+ 2*t),Gamma[t + \[Mu]+ 1]*Gamma[t + \[Nu]+ 1]*Gamma[t + \[Mu]+ \[Nu]+ 1]], {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
10.9.E30 J ν 2 ⁡ ( z ) + Y ν 2 ⁡ ( z ) = 8 π 2 ⁢ ∫ 0 ∞ cosh ⁡ ( 2 ⁢ ν ⁢ t ) ⁢ K 0 ⁡ ( 2 ⁢ z ⁢ sinh ⁡ t ) ⁢ d t Bessel-J 𝜈 2 𝑧 Bessel-Y-Weber 𝜈 2 𝑧 8 superscript 𝜋 2 superscript subscript 0 2 𝜈 𝑡 modified-Bessel-second-kind 0 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle{J_{\nu}^{2}}\left(z\right)+{Y_{\nu}^{2}}\left(z% \right)=\frac{8}{\pi^{2}}\int_{0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2% z\sinh t\right)\mathrm{d}t}}
\BesselJ{\nu}^{2}@{z}+\BesselY{\nu}^{2}@{z} = \frac{8}{\pi^{2}}\int_{0}^{\infty}\cosh@{2\nu t}\modBesselK{0}@{2z\sinh@@{t}}\diff{t}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>% 0,\Re((-\nu)+k+1)>0}} 
(BesselJ(nu, z))^(2)+ (BesselY(nu, z))^(2) = (8)/((Pi)^(2))*int(cosh(2*nu*t)*BesselK(0, 2*z*sinh(t)), t = 0..infinity)
 
(BesselJ[\[Nu], z])^(2)+ (BesselY[\[Nu], z])^(2) == Divide[8,(Pi)^(2)]*Integrate[Cosh[2*\[Nu]*t]*BesselK[0, 2*z*Sinh[t]], {t, 0, Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
Retrieved from "https://lct.wmflabs.org/w/index.php?title=10.9&oldid=58185"