13.23

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
13.23.E1	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu+\nu+\frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+\nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right)}}$ \int_{0}^{\infty}e^{-zt}t^{\nu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\mu+\nu+\tfrac{1}{2}}}{\left(z+\frac{1}{2}\right)^{\mu+\nu+\frac{1}{2}}}\*\genhyperF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+\nu}{1+2\mu}{\frac{1}{z+\frac{1}{2}}}	${\displaystyle{\displaystyle\Re\mu+\nu+\tfrac{1}{2}>0,\Re z>\tfrac{1}{2},\Re(\mu+\nu+\tfrac{1}{2})>0}}$	int(exp(- z*t)*(t)^(nu - 1)* WhittakerM(kappa, mu, t), t = 0..infinity) = (GAMMA(mu + nu +(1)/(2)))/((z +(1)/(2))^(mu + nu +(1)/(2)))* hypergeom([(1)/(2)+ mu - kappa ,(1)/(2)+ mu + nu], [1 + 2*mu], (1)/(z +(1)/(2)))	Integrate[Exp[- z*t]*(t)^(\[Nu]- 1)* WhittakerM[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Mu]+ \[Nu]+Divide[1,2]],(z +Divide[1,2])^(\[Mu]+ \[Nu]+Divide[1,2])]* HypergeometricPFQ[{Divide[1,2]+ \[Mu]- \[Kappa],Divide[1,2]+ \[Mu]+ \[Nu]}, {1 + 2*\[Mu]}, Divide[1,z +Divide[1,2]]]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.23.E2	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\Gamma\left(2\mu+1\right)\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1}{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}}}}$ \int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \EulerGamma@{2\mu+1}\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1}{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}}	${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re z>\tfrac{1}{2},\Re(2\mu+1)>0}}$	int(exp(- z*t)*(t)^(mu -(1)/(2))* WhittakerM(kappa, mu, t), t = 0..infinity) = GAMMA(2*mu + 1)*(z +(1)/(2))^(- kappa - mu -(1)/(2))*(z -(1)/(2))^(kappa - mu -(1)/(2))	Integrate[Exp[- z*t]*(t)^(\[Mu]-Divide[1,2])* WhittakerM[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[2*\[Mu]+ 1]*(z +Divide[1,2])^(- \[Kappa]- \[Mu]-Divide[1,2])*(z -Divide[1,2])^(\[Kappa]- \[Mu]-Divide[1,2])	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.23.E3	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}\right)\Gamma\left(\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right)\Gamma\left(\frac{1}{2}+\mu-\nu\right)}}}$ \frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\mu+\nu+\frac{1}{2}}\EulerGamma@{\kappa-\nu}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}\EulerGamma@{\frac{1}{2}+\mu-\nu}}	${\displaystyle{\displaystyle-\tfrac{1}{2}-\Re\mu<\Re\nu,\Re\nu<\Re\kappa,\Re(1+2\mu)>0,\Re(\mu+\nu+\frac{1}{2})>0,\Re(\kappa-\nu)>0,\Re(\frac{1}{2}+\mu+\kappa)>0,\Re(\frac{1}{2}+\mu-\nu)>0}}$	(1)/(GAMMA(1 + 2*mu))*int(exp(-(1)/(2)*t)*(t)^(nu - 1)* WhittakerM(kappa, mu, t), t = 0..infinity) = (GAMMA(mu + nu +(1)/(2))*GAMMA(kappa - nu))/(GAMMA((1)/(2)+ mu + kappa)*GAMMA((1)/(2)+ mu - nu))	Divide[1,Gamma[1 + 2*\[Mu]]]*Integrate[Exp[-Divide[1,2]*t]*(t)^(\[Nu]- 1)* WhittakerM[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Mu]+ \[Nu]+Divide[1,2]]*Gamma[\[Kappa]- \[Nu]],Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- \[Nu]]]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.23.E4	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\mathrm{d}t=\Gamma\left(\tfrac{1}{2}+\mu+\nu\right)\Gamma\left(\tfrac{1}{2}-\mu+\nu\right)\*{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu\atop\nu-\kappa+1};\tfrac{1}{2}-z\right)}}$ \int_{0}^{\infty}e^{-zt}t^{\nu-1}\WhittakerconfhyperW{\kappa}{\mu}@{t}\diff{t} = \EulerGamma@{\tfrac{1}{2}+\mu+\nu}\EulerGamma@{\tfrac{1}{2}-\mu+\nu}\*\genhyperOlverF{2}{1}@@{\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu}{\nu-\kappa+1}{\tfrac{1}{2}-z}	${\displaystyle{\displaystyle\Re\left(\nu+\tfrac{1}{2}\right)>|\Re\mu|,\Re z>-\tfrac{1}{2},\Re(\tfrac{1}{2}+\mu+\nu)>0,\Re(\tfrac{1}{2}-\mu+\nu)>0}}$	int(exp(- z*t)*(t)^(nu - 1)* WhittakerW(kappa, mu, t), t = 0..infinity) = GAMMA((1)/(2)+ mu + nu)*GAMMA((1)/(2)- mu + nu)* hypergeom([(1)/(2)- mu + nu ,(1)/(2)+ mu + nu], [nu - kappa + 1], (1)/(2)- z)	Integrate[Exp[- z*t]*(t)^(\[Nu]- 1)* WhittakerW[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[Divide[1,2]+ \[Mu]+ \[Nu]]*Gamma[Divide[1,2]- \[Mu]+ \[Nu]]* HypergeometricPFQRegularized[{Divide[1,2]- \[Mu]+ \[Nu],Divide[1,2]+ \[Mu]+ \[Nu]}, {\[Nu]- \[Kappa]+ 1}, Divide[1,2]- z]	Failure	Aborted	Failed [276 / 300] Result: Float(infinity)+Float(infinity)*I Test Values: {kappa = 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, z = 1/2} Result: .2394973555+.5504747838e-1*I Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2-1/2*I*3^(1/2), z = 1/2*3^(1/2)+1/2*I} ... skip entries to safe data	Skipped - Because timed out
13.23.E5	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\frac{1}{2}+\mu+\nu\right)\Gamma\left(\frac{1}{2}-\mu+\nu\right)\Gamma\left(-\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}}}$ \int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}\WhittakerconfhyperW{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\frac{1}{2}+\mu+\nu}\EulerGamma@{\frac{1}{2}-\mu+\nu}\EulerGamma@{-\kappa-\nu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}	${\displaystyle{\displaystyle|\Re\mu|-\tfrac{1}{2}<\Re\nu,\Re\nu<-\Re\kappa,\Re(\frac{1}{2}+\mu+\nu)>0,\Re(\frac{1}{2}-\mu+\nu)>0,\Re(-\kappa-\nu)>0,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re(\frac{1}{2}-\mu-\kappa)>0}}$	int(exp((1)/(2)*t)*(t)^(nu - 1)* WhittakerW(kappa, mu, t), t = 0..infinity) = (GAMMA((1)/(2)+ mu + nu)*GAMMA((1)/(2)- mu + nu)*GAMMA(- kappa - nu))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))	Integrate[Exp[Divide[1,2]*t]*(t)^(\[Nu]- 1)* WhittakerW[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1,2]+ \[Mu]+ \[Nu]]*Gamma[Divide[1,2]- \[Mu]+ \[Nu]]*Gamma[- \[Kappa]- \[Nu]],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]	Failure	Aborted	Manual Skip!	Successful [Tested: 56]
13.23.E6	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac{z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}I_{2\mu}\left(2\sqrt{z}\right)}}$ \frac{1}{\EulerGamma@{1+2\mu}2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}\WhittakerconfhyperM{\kappa}{\mu}@{t^{-1}}\diff{t} = \frac{z^{-\kappa-\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\modBesselI{2\mu}@{2\sqrt{z}}	${\displaystyle{\displaystyle\Re z>0,\Re(1+2\mu)>0,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re((2\mu)+k+1)>0}}$	(1)/(GAMMA(1 + 2*mu)*2*Pi*I)*int(exp(z*t +(1)/(2)*(t)^(- 1))*(t)^(kappa)* WhittakerM(kappa, mu, (t)^(- 1)), t = - infinity..(0 +)) = ((z)^(- kappa -(1)/(2)))/(GAMMA((1)/(2)+ mu - kappa))*BesselI(2*mu, 2*sqrt(z))	Divide[1,Gamma[1 + 2*\[Mu]]*2*Pi*I]*Integrate[Exp[z*t +Divide[1,2]*(t)^(- 1)]*(t)^\[Kappa]* WhittakerM[\[Kappa], \[Mu], (t)^(- 1)], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[(z)^(- \[Kappa]-Divide[1,2]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]*BesselI[2*\[Mu], 2*Sqrt[z]]	Error	Failure	-	Error
13.23.E7	${\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}W_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac{2z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}K_{2\mu}\left(2\sqrt{z}\right)}}$ \frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}\WhittakerconfhyperW{\kappa}{\mu}@{t^{-1}}\diff{t} = \frac{2z^{-\kappa-\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\modBesselK{2\mu}@{2\sqrt{z}}	${\displaystyle{\displaystyle\Re z>0,\Re(\frac{1}{2}+\mu-\kappa)>0,\Re(\frac{1}{2}-\mu-\kappa)>0}}$	(1)/(2*Pi*I)*int(exp(z*t +(1)/(2)*(t)^(- 1))*(t)^(kappa)* WhittakerW(kappa, mu, (t)^(- 1)), t = - infinity..(0 +)) = (2*(z)^(- kappa -(1)/(2)))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))*BesselK(2*mu, 2*sqrt(z))	Divide[1,2*Pi*I]*Integrate[Exp[z*t +Divide[1,2]*(t)^(- 1)]*(t)^\[Kappa]* WhittakerW[\[Kappa], \[Mu], (t)^(- 1)], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[2*(z)^(- \[Kappa]-Divide[1,2]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]*BesselK[2*\[Mu], 2*Sqrt[z]]	Error	Failure	-	Error
13.23.E8	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}\cos\left(2xt\right)e^{-\frac{1}{2}t^{2}}t^{-2\mu-1}M_{\kappa,\mu}\left(t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\Gamma\left(\frac{1}{2}+\mu+\kappa\right)}W_{\frac{1}{2}\kappa-\frac{3}{2}\mu,\frac{1}{2}\kappa+\frac{1}{2}\mu}\left(x^{2}\right)}}$ \frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}\cos@{2xt}e^{-\frac{1}{2}t^{2}}t^{-2\mu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t^{2}}\diff{t} = \frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\WhittakerconfhyperW{\frac{1}{2}\kappa-\frac{3}{2}\mu}{\frac{1}{2}\kappa+\frac{1}{2}\mu}@{x^{2}}	${\displaystyle{\displaystyle\Re\left(\kappa+\mu\right)>-\tfrac{1}{2},\Re(1+2\mu)>0,\Re(\frac{1}{2}+\mu+\kappa)>0}}$	(1)/(GAMMA(1 + 2*mu))*int(cos(2*x*t)*exp(-(1)/(2)*(t)^(2))*(t)^(- 2*mu - 1)* WhittakerM(kappa, mu, (t)^(2)), t = 0..infinity) = (sqrt(Pi)*exp(-(1)/(2)*(x)^(2))*(x)^(mu + kappa - 1))/(2*GAMMA((1)/(2)+ mu + kappa))*WhittakerW((1)/(2)*kappa -(3)/(2)*mu, (1)/(2)*kappa +(1)/(2)*mu, (x)^(2))	Divide[1,Gamma[1 + 2*\[Mu]]]*Integrate[Cos[2*x*t]*Exp[-Divide[1,2]*(t)^(2)]*(t)^(- 2*\[Mu]- 1)* WhittakerM[\[Kappa], \[Mu], (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi]*Exp[-Divide[1,2]*(x)^(2)]*(x)^(\[Mu]+ \[Kappa]- 1),2*Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]*WhittakerW[Divide[1,2]*\[Kappa]-Divide[3,2]*\[Mu], Divide[1,2]*\[Kappa]+Divide[1,2]*\[Mu], (x)^(2)]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.23.E9	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}M_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa+\nu\right)}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa+3\mu-\nu+\frac{1}{2}),\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}\left(x\right)}}$ \int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}\WhittakerconfhyperM{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{1+2\mu}}{\EulerGamma@{\frac{1}{2}-\mu+\kappa+\nu}}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*\WhittakerconfhyperM{\frac{1}{2}(\kappa+3\mu-\nu+\frac{1}{2})}{\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}@{x}	${\displaystyle{\displaystyle x>0,-\tfrac{1}{2}<\Re\mu,\Re\mu<\Re\left(\kappa+\tfrac{1}{2}\nu\right)+\tfrac{3}{4},\Re(\nu+k+1)>0,\Re(1+2\mu)>0,\Re(\frac{1}{2}-\mu+\kappa+\nu)>0}}$	int(exp(-(1)/(2)*t)*(t)^(mu -(1)/(2)*(nu + 1))* WhittakerM(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(1 + 2*mu))/(GAMMA((1)/(2)- mu + kappa + nu))* exp(-(1)/(2)*x)*(x)^((1)/(2)*(kappa - mu -(3)/(2)))* WhittakerM((1)/(2)*(kappa + 3*mu - nu +(1)/(2)), (1)/(2)*(kappa - mu + nu -(1)/(2)), x)	Integrate[Exp[-Divide[1,2]*t]*(t)^(\[Mu]-Divide[1,2]*(\[Nu]+ 1))* WhittakerM[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[1 + 2*\[Mu]],Gamma[Divide[1,2]- \[Mu]+ \[Kappa]+ \[Nu]]]* Exp[-Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Kappa]- \[Mu]-Divide[3,2]))* WhittakerM[Divide[1,2]*(\[Kappa]+ 3*\[Mu]- \[Nu]+Divide[1,2]), Divide[1,2]*(\[Kappa]- \[Mu]+ \[Nu]-Divide[1,2]), x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.23.E10	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}M_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right)}\*W_{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2}),\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}\left(x\right)}}$ \frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperM{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\WhittakerconfhyperW{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2})}{\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}@{x}	${\displaystyle{\displaystyle x>0,-1<\Re\nu,\Re\nu<2\Re\left(\mu+\kappa\right)+\tfrac{1}{2},\Re(\nu+k+1)>0,\Re(1+2\mu)>0,\Re(\frac{1}{2}+\mu+\kappa)>0}}$	(1)/(GAMMA(1 + 2*mu))*int(exp(-(1)/(2)*t)*(t)^((1)/(2)*(nu - 1)- mu)* WhittakerM(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (exp(-(1)/(2)*x)*(x)^((1)/(2)*(kappa + mu -(3)/(2))))/(GAMMA((1)/(2)+ mu + kappa))* WhittakerW((1)/(2)*(kappa - 3*mu + nu +(1)/(2)), (1)/(2)*(kappa + mu - nu -(1)/(2)), x)	Divide[1,Gamma[1 + 2*\[Mu]]]*Integrate[Exp[-Divide[1,2]*t]*(t)^(Divide[1,2]*(\[Nu]- 1)- \[Mu])* WhittakerM[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Exp[-Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Kappa]+ \[Mu]-Divide[3,2])),Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]* WhittakerW[Divide[1,2]*(\[Kappa]- 3*\[Mu]+ \[Nu]+Divide[1,2]), Divide[1,2]*(\[Kappa]+ \[Mu]- \[Nu]-Divide[1,2]), x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
13.23.E11	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-2\mu+1\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*e^{\frac{1}{2}x}x^{\frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*W_{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{2}),\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}\left(x\right)}}$ \int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperW{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-2\mu+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*e^{\frac{1}{2}x}x^{\frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*\WhittakerconfhyperW{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{2})}{\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}@{x}	${\displaystyle{\displaystyle x>0,\max(2\Re\mu-1<\Re\nu,-1)<\Re\nu,\Re\nu<2\Re\mu-\kappa+\tfrac{3}{2},\Re(\nu+k+1)>0,\Re(\nu-2\mu+1)>0,\Re(\frac{1}{2}+\mu-\kappa)>0}}$	int(exp((1)/(2)*t)*(t)^((1)/(2)*(nu - 1)- mu)* WhittakerW(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(nu - 2*mu + 1))/(GAMMA((1)/(2)+ mu - kappa))* exp((1)/(2)*x)*(x)^((1)/(2)*(mu - kappa -(3)/(2)))* WhittakerW((1)/(2)*(kappa + 3*mu - nu -(1)/(2)), (1)/(2)*(kappa - mu + nu +(1)/(2)), x)	Integrate[Exp[Divide[1,2]*t]*(t)^(Divide[1,2]*(\[Nu]- 1)- \[Mu])* WhittakerW[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- 2*\[Mu]+ 1],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Exp[Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Mu]- \[Kappa]-Divide[3,2]))* WhittakerW[Divide[1,2]*(\[Kappa]+ 3*\[Mu]- \[Nu]-Divide[1,2]), Divide[1,2]*(\[Kappa]- \[Mu]+ \[Nu]+Divide[1,2]), x]	Failure	Aborted	Manual Skip!	Skipped - Because timed out
13.23.E12	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-2\mu+1\right)}{\Gamma\left(\frac{3}{2}-\mu-\kappa+\nu\right)}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2}),\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}\left(x\right)}}$ \int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperW{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-2\mu+1}}{\EulerGamma@{\frac{3}{2}-\mu-\kappa+\nu}}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*\WhittakerconfhyperM{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2})}{\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}@{x}	${\displaystyle{\displaystyle x>0,\max(2\Re\mu-1<\Re\nu,-1)<\Re\nu,\Re(\nu+k+1)>0,\Re(\nu-2\mu+1)>0,\Re(\frac{3}{2}-\mu-\kappa+\nu)>0}}$	int(exp(-(1)/(2)*t)*(t)^((1)/(2)*(nu - 1)- mu)* WhittakerW(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(nu - 2*mu + 1))/(GAMMA((3)/(2)- mu - kappa + nu))* exp(-(1)/(2)*x)*(x)^((1)/(2)*(mu + kappa -(3)/(2)))* WhittakerM((1)/(2)*(kappa - 3*mu + nu +(1)/(2)), (1)/(2)*(nu - mu - kappa +(1)/(2)), x)	Integrate[Exp[-Divide[1,2]*t]*(t)^(Divide[1,2]*(\[Nu]- 1)- \[Mu])* WhittakerW[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- 2*\[Mu]+ 1],Gamma[Divide[3,2]- \[Mu]- \[Kappa]+ \[Nu]]]* Exp[-Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Mu]+ \[Kappa]-Divide[3,2]))* WhittakerM[Divide[1,2]*(\[Kappa]- 3*\[Mu]+ \[Nu]+Divide[1,2]), Divide[1,2]*(\[Nu]- \[Mu]- \[Kappa]+Divide[1,2]), x]	Failure	Aborted	Manual Skip!	Skipped - Because timed out