Mehler-Dirichlet Integral

Author

Eric W. Weisstein
November 29, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Mehler-DirichletIntegral.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Formula

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

Numerical Check

With[{n=1.2,α=1.},((Sqrt[2])/π)NIntegrate[((Cos[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[α]])),{ϕ,0,α}]]

LegendreP[1.2,Cos[1.]]

Subtract@@@Table[With[{n=RandomComplexDisk[],α=RandomComplexDisk[]},{LegendreP[n,Cos[α]],((Sqrt[2])/π)NIntegrate[((Cos[(n+(1/2))ϕ])/(Sqrt[Cos[ϕ]-Cos[α]])),{ϕ,0,α}]}],{100}]//Chop[#,10]&//Union