10.9

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.9.E1	${\displaystyle{\displaystyle J_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta\right)\mathrm{d}\theta}}$ \BesselJ{0}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta}	${\displaystyle{\displaystyle\Re(0+k+1)>0}}$	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	${\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta\right)\mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\cos\theta\right)\mathrm{d}\theta}}$ \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}}\diff{\theta} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}\diff{\theta}	${\displaystyle{\displaystyle\Re(0+k+1)>0}}$	(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	${\displaystyle{\displaystyle J_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-n\theta\right)\mathrm{d}\theta}}$ \BesselJ{n}@{z} = \frac{1}{\pi}\int_{0}^{\pi}\cos@{z\sin@@{\theta}-n\theta}\diff{\theta}	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	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	${\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-n\theta\right)\mathrm{d}\theta=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos\theta}\cos\left(n\theta\right)\mathrm{d}\theta}}$ \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}	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	(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	${\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(z\cos\theta\right)\left(\gamma+\ln\left(2z{\sin^{2}}\theta\right)\right)\mathrm{d}\theta}}$ \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}	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta}}$ \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}	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu+\tfrac{1}{2})>0}}$	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	${\displaystyle{\displaystyle\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\cos\left(zt\right)\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\nu+\tfrac{1}{2})>0}}$	(((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	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t\right)}}$ \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)	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(\nu+\tfrac{1}{2})>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-z\sinh t-\nu t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t}+e^{-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(x\cosh t-\tfrac{1}{2}\nu\pi\right)\cosh\left(\nu t\right)\mathrm{d}t}}$ \BesselJ{\nu}@{x} = \frac{2}{\pi}\int_{0}^{\infty}\sin@{x\cosh@@{t}-\tfrac{1}{2}\nu\pi}\cosh@{\nu t}\diff{t}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle Y_{\nu}\left(x\right)=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t-\tfrac{1}{2}\nu\pi\right)\cosh\left(\nu t\right)\mathrm{d}t}}$ \BesselY{\nu}@{x} = -\frac{2}{\pi}\int_{0}^{\infty}\cos@{x\cosh@@{t}-\tfrac{1}{2}\nu\pi}\cosh@{\nu t}\diff{t}	${\displaystyle{\displaystyle|\Re\nu|<1,x>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{0}\left(x\right)=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(x\cosh t\right)\mathrm{d}t}}$ \BesselJ{0}@{x} = \frac{2}{\pi}\int_{0}^{\infty}\sin@{x\cosh@@{t}}\diff{t}	${\displaystyle{\displaystyle x>0,\Re(0+k+1)>0}}$	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	${\displaystyle{\displaystyle Y_{0}\left(x\right)=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t\right)\mathrm{d}t}}$ \BesselY{0}@{x} = -\frac{2}{\pi}\int_{0}^{\infty}\cos@{x\cosh@@{t}}\diff{t}	${\displaystyle{\displaystyle x>0,\Re(0+k+1)>0,\Re((-0)+k+1)>0}}$	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	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{e^{-\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t-\nu t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle 0<\operatorname{ph}z,\operatorname{ph}z<\pi}}$	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	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-\nu t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}\frac{\sin\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}}}}$ \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}}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}-\nu)>0}}$	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	${\displaystyle{\displaystyle Y_{\nu}\left(x\right)=-\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}\frac{\cos\left(xt\right)\mathrm{d}t}{(t^{2}-1)^{\nu+\frac{1}{2}}}}}$ \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}}}	${\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2},x>0,\Re(\tfrac{1}{2}-\nu)>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	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	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}J_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t-\nu t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Re\left(z+\zeta\right)>0,\Re(\nu+k+1)>0}}$	((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	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}Y_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t+\zeta\cosh t}+e^{-\nu t-\zeta\cosh t}\cos\left(\nu\pi\right)\right)\*e^{-z\sinh t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Re\left(z+\zeta\right)>0,\Re\left(z-\zeta\right)>0,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	((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	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(1)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi i}e^{-\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t+i\zeta\sinh t-\nu t}\mathrm{d}t}}$ \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	${\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(2)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i}e^{\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Im\left(z+\zeta\right)<0,\Im\left(z-\zeta\right)<0}}$	((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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}$ \BesselJ{\nu}@{z} = \frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z\sinh@@{t}-\nu t}\diff{t}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{1}{\pi i}\int_{-\infty}^{\infty+\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}$ \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	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{1}{\pi i}\int_{-\infty}^{\infty-\pi i}e^{z\sinh t-\nu t}\mathrm{d}t}}$ \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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{2\pi i}\int_{-\infty}^{(0+)}\exp\left(t-\frac{z^{2}}{4t}\right)\frac{\mathrm{d}t}{t^{\nu+1}}}}$ \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}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{2}-\nu\right)(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left(zt\right)(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(\frac{1}{2}-\nu)>0}}$	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	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1+i\infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2}<\tfrac{1}{2}\pi,\ldots<\tfrac{1}{2}\pi,|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\tfrac{1}{2}-\nu)>0}}$	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	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1-i\infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2}<\tfrac{1}{2}\pi,\ldots<\tfrac{1}{2}\pi,|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\tfrac{1}{2}-\nu)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(-t\right)(\tfrac{1}{2}x)^{\nu+2t}}{\Gamma\left(\nu+t+1\right)}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Re\nu>0,x>0,\Re(\nu+k+1)>0,\Re(-t)>0,\Re(\nu+t+1)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{-\infty-ic}^{-\infty+ic}\frac{\Gamma\left(t\right)}{\Gamma\left(\nu-t+1\right)}(\tfrac{1}{2}z)^{\nu-2t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re t>0,\Re(\nu-t+1)>0}}$	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	${\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=-\frac{e^{-\frac{1}{2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle 0<\operatorname{ph}z,\operatorname{ph}z<\pi,\Re t>0,\Re(t-\nu)>0}}$	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	${\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=\frac{e^{\frac{1}{2}\nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<0,\Re t>0,\Re(t-\nu)>0}}$	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	${\displaystyle{\displaystyle J_{\mu}\left(z\right)J_{\nu}\left(z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}J_{\mu+\nu}\left(2z\cos\theta\right)\cos\left((\mu-\nu)\theta\right)\mathrm{d}\theta}}$ \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}	${\displaystyle{\displaystyle\Re\left(\mu+\nu\right)>-1,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,\Re((\mu+\nu)+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=\frac{2}{\pi}\int_{0}^{\pi/2}J_{2\nu}\left(2(z\zeta)^{\frac{1}{2}}\sin\theta\right)\cos\left((z-\zeta)\cos\theta\right)\mathrm{d}\theta}}$ \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}	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((2\nu)+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\*\exp\left(\frac{1}{2}t-\frac{z^{2}+\zeta^{2}}{2t}\right)I_{\nu}\left(\frac{z\zeta}{t}\right)\frac{\mathrm{d}t}{t}}}$ \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}	${\displaystyle{\displaystyle\Re\nu>-1,\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle J_{\mu}\left(x\right)J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+\mu+\nu+1\right)(\tfrac{1}{2}x)^{\mu+\nu+2t}}{\Gamma\left(t+\mu+1\right)\Gamma\left(t+\nu+1\right)\Gamma\left(t+\mu+\nu+1\right)}\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle x>0,\Re((\mu)+k+1)>0,\Re(\nu+k+1)>0,\Re(-t)>0,\Re(2t+\mu+\nu+1)>0,\Re(t+\mu+1)>0,\Re(t+\nu+1)>0,\Re(t+\mu+\nu+1)>0}}$	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	${\displaystyle{\displaystyle{J_{\nu}^{2}}\left(z\right)+{Y_{\nu}^{2}}\left(z\right)=\frac{8}{\pi^{2}}\int_{0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2z\sinh t\right)\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	(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