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Laplace's Integral

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Laplace's integral is one of the following integral representations of the Legendre polynomial P_n(x),

P_n(x)=1/piint_0^pi(du)/((x+sqrt(x^2-1)cosu)^(n+1))du
(1)
=1/piint_0^pi(x+sqrt(x^2-1)cosu)^ndu.
(2)

It can be evaluated in terms of the hypergeometric function.

See also

Legendre Polynomial, Laplace-Mehler Integral, Mehler-Dirichlet Integral

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Cite this as:

Weisstein, Eric W. "Laplace's Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplacesIntegral.html

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