Legendre Function - Integral Representations

Integral Representations

The Legendre functions can be written as contour integrals. For example

$P_\lambda(z) =\frac{1}{2\pi i} \int_{1,z} \frac{(t^2-1)^\lambda}{2^\lambda(t-z)^{\lambda+1}}dt$

where the contour circles around the points 1 and z in the positive direction and does not circle around −1. For real x, we have

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