18.10

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.10.E1	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}}}$ \frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}} = \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi,\alpha>-\tfrac{1}{2}}}$	(JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1)) = (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1))	Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]] == Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 27]
18.10.E1	${\displaystyle{\displaystyle\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\mathrm{d}\phi}}$ \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}} = \frac{2^{\alpha+\frac{1}{2}}\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma@{\alpha+\frac{1}{2}}}(\sin@@{\theta})^{-2\alpha}\int_{0}^{\theta}\frac{\cos@{(n+\alpha+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{-\alpha+\frac{1}{2}}}\diff{\phi}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi,\alpha>-\tfrac{1}{2},\Re(\alpha+1)>0,\Re(\alpha+\frac{1}{2})>0}}$	(GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) = ((2)^(alpha +(1)/(2))* GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*(sin(theta))^(- 2*alpha)* int((cos((n + alpha +(1)/(2))*phi))/((cos(phi)- cos(theta))^(- alpha +(1)/(2))), phi = 0..theta)	Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] == Divide[(2)^(\[Alpha]+Divide[1,2])* Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*(Sin[\[Theta]])^(- 2*\[Alpha])* Integrate[Divide[Cos[(n + \[Alpha]+Divide[1,2])*\[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(- \[Alpha]+Divide[1,2])], {\[Phi], 0, \[Theta]}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 27]	Skipped - Because timed out
18.10.E2	${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos\left((n+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{\frac{1}{2}}}\mathrm{d}\phi}}$ \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos@{(n+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{\frac{1}{2}}}\diff{\phi}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi}}$	LegendreP(n, cos(theta)) = ((2)^((1)/(2)))/(Pi)*int((cos((n +(1)/(2))*phi))/((cos(phi)- cos(theta))^((1)/(2))), phi = 0..theta)	LegendreP[n, Cos[\[Theta]]] == Divide[(2)^(Divide[1,2]),Pi]*Integrate[Divide[Cos[(n +Divide[1,2])*\[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(Divide[1,2])], {\[Phi], 0, \[Theta]}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
18.10.E4	${\displaystyle{\displaystyle{\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\mathrm{d}\phi}}}$ {\frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}}=\frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}} = \frac{\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\*(\sin@@{\phi})^{2\alpha}\diff{\phi}}	${\displaystyle{\displaystyle\alpha>-\frac{1}{2},\Re(\alpha+1)>0}}$	(JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1)) = (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) = (GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*int((cos(theta)+ I*sin(theta)*cos(phi))^(n)*(sin(phi))^(2*alpha), phi = 0..Pi)	Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]] == Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] == Divide[Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n)*(Sin[\[Phi]])^(2*\[Alpha]), {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
18.10.E5	${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\mathrm{d}\phi}}$ \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{1}{\pi}\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\diff{\phi}		LegendreP(n, cos(theta)) = (1)/(Pi)*int((cos(theta)+ I*sin(theta)*cos(phi))^(n), phi = 0..Pi)	LegendreP[n, Cos[\[Theta]]] == Divide[1,Pi]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n), {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 30]	Skipped - Because timed out
18.10.E7	${\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\mathrm{d}t}}$ \HermitepolyH{n}@{x} = \frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\diff{t}		HermiteH(n, x) = ((2)^(n))/((Pi)^((1)/(2)))*int((x + I*t)^(n)* exp(- (t)^(2)), t = - infinity..infinity)	HermiteH[n, x] == Divide[(2)^(n),(Pi)^(Divide[1,2])]*Integrate[(x + I*t)^(n)* Exp[- (t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 9]	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
18.10.E9	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}J_{\alpha}\left(2\sqrt{xt}\right)\mathrm{d}t}}$ \LaguerrepolyL[\alpha]{n}@{x} = \frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}\BesselJ{\alpha}@{2\sqrt{xt}}\diff{t}	${\displaystyle{\displaystyle\alpha>-1,\Re((\alpha)+k+1)>0}}$	LaguerreL(n, alpha, x) = (exp(x)*(x)^(-(1)/(2)*alpha))/(factorial(n))*int(exp(- t)*(t)^(n +(1)/(2)*alpha)* BesselJ(alpha, 2*sqrt(x*t)), t = 0..infinity)	LaguerreL[n, \[Alpha], x] == Divide[Exp[x]*(x)^(-Divide[1,2]*\[Alpha]),(n)!]*Integrate[Exp[- t]*(t)^(n +Divide[1,2]*\[Alpha])* BesselJ[\[Alpha], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
18.10.E10	${\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t}}$ \HermitepolyH{n}@{x} = \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t}		HermiteH(n, x) = ((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity)	HermiteH[n, x] == Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.10.E10	${\displaystyle{\displaystyle\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t=\frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi\right)\mathrm{d}t}}$ \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t} = \frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos@{2xt-\tfrac{1}{2}n\pi}\diff{t}		((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity) = ((2)^(n + 1))/((Pi)^((1)/(2)))*exp((x)^(2))*int(exp(- (t)^(2))*(t)^(n)* cos(2*x*t -(1)/(2)*n*Pi), t = 0..infinity)	Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(2)^(n + 1),(Pi)^(Divide[1,2])]*Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Cos[2*x*t -Divide[1,2]*n*Pi], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Successful [Tested: 9]