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Laplace-Mehler Integral


The Lehmer-Mahler is the following integral representation for the Legendre polynomial P_n(x):

P_n(costheta)=1/piint_0^pi(costheta+isinthetacosphi)^ndphi
(1)
=(sqrt(2))/piint_0^theta(cos[(n+1/2)phi])/(sqrt(cosphi-costheta))dphi
(2)
=(sqrt(2))/piint_theta^pi(sin[(n+1/2)phi])/(sqrt(costheta-cosphi))dphi.
(3)

See also

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

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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.

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Laplace-Mehler Integral

Cite this as:

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

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