Laplace-Mehler Integral

Author

Eric W. Weisstein
March 21, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/Laplace-MehlerIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Laplace-MehlerIntegral.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Evaluation

(1/π)(∫_0^π)(((cos(θ)+i sin(θ) cos(ϕ)))^n)dϕ

v5.2

(expr=(1/π)Integrate[(((Cos[θ]+I Sin[θ]Cos[ϕ]))^n),{ϕ,0,2π},GenerateConditions->False]//FullSimplify)//Timing

Table[(1/(Gamma[(3/2)+n]))(2 (Sqrt[(1/(1+I Cot[θ]))]) (Sqrt[(1/(π-I π Cot[θ]))]) Csc[θ] Gamma[1+n] (-I ((((E^(I θ))))^(1+n)) Hypergeometric2F1[(1/2),1+n,(3/2)+n,(E^(2 I θ))]+I ((((E^(-I θ))))^(1+n)) Hypergeometric2F1[(1/2),1+n,(3/2)+n,Cos[2 θ]-I Sin[2 θ]]))/2-LegendreP[n,Cos[θ]]/.{n->Random[Integer,{0,1}20],θ->Random[Real,{0,2π},20]},{20}]//Chop

V7

(expr=(1/π)Integrate[(((Cos[θ]+I Sin[θ]Cos[ϕ]))^n),{ϕ,0,π},GenerateConditions->False]//FullSimplify)//Timing

Table[expr-LegendreP[n,Cos[θ]]/.{n->RandomInteger[{0,20}],θ->RandomReal[{0,2π}]},{10}]//Chop//Union

(((Sqrt[2]) )/π)(∫_0^θ)((cos((n+(1/2)) ϕ))/(Sqrt[cos(ϕ)-cos(θ)]))dϕ

V5.2

(1/π)Integrate[((Sin[(n+(1/2))ϕ])/(Sqrt[Cos[θ]-Cos[ϕ]])),{ϕ,θ,π}]//Timing

V7

((Sqrt[2])/π)Integrate[((Cos[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[θ]])),{ϕ,0,θ}]//Timing

With[{n=10,θ=.2},Plot[((Cos[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[θ]])),{ϕ,0,θ}]]

Subtract@@@Table[With[{n=RandomInteger[{0,20}],θ=RandomReal[{0,π},WorkingPrecision->50]},{((Sqrt[2])/π)NIntegrate[((Cos[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[θ]])),{ϕ,0,θ},Method->"DoubleExponential",WorkingPrecision->50,MaxRecursion->20],LegendreP[n,Cos[θ]]}],{20}]//Chop//Union//Quiet//N

(((Sqrt[2]) )/π)(∫_θ^π)((sin((n+(1/2)) ϕ))/(Sqrt[cos(ϕ)-cos(θ)]))dϕ

V5.2

((Sqrt[2])/π)Integrate[((Sin[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[θ]])),{ϕ,θ,π}]//Timing

V7

((Sqrt[2])/π)Integrate[((Sin[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[θ]])),{ϕ,θ,π},Assumptions->0<θ<π]//Timing

Subtract@@@Table[With[{n=RandomInteger[{0,20}],θ=RandomReal[{0,π},WorkingPrecision->50]},{((Sqrt[2])/π)NIntegrate[((Sin[(n+(1/2))ϕ])/(Sqrt[Cos[θ]-Cos[ϕ]])),{ϕ,θ,π},Method->"DoubleExponential",WorkingPrecision->50,MaxRecursion->20],LegendreP[n,Cos[θ]]}],{10}]//Chop//Union//Quiet//N