Laplace-Mehler Integral
The Lehmer-Mahler is the following integral representation for the Legendre polynomial P_n(x):
![画像:(sqrt(2))/piint_0^theta(cos[(n+1/2)phi])/(sqrt(cosphi-costheta))dphi](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Laplace-MehlerIntegral/Inline7.svg){kind=link}
![画像:(sqrt(2))/piint_theta^pi(sin[(n+1/2)phi])/(sqrt(costheta-cosphi))dphi.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Laplace-MehlerIntegral/Inline10.svg){kind=link}
See also
Laplace's Integral, Legendre Polynomial, Mehler-Dirichlet IntegralExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea, 1955.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1463, 1980.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Referenced on Wolfram|Alpha
Laplace-Mehler IntegralCite this as:
Weisstein, Eric W. "Laplace-Mehler Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Laplace-MehlerIntegral.html