Complex Analysis: Theory and Applications · Problem 6.5.19

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · cauchy-integral-formula

Problem

The Legendre polynomial

P_n(z)

is defined by

P_n(z) = \frac{1}{2^n n!}\frac{d^n}{dz^n}[(z^2-1)^n].

Use Cauchy's integral formula to show that

P_n(z) = \frac{1}{2\pi i}\int_C \frac{(\xi^2-1)^n}{2^n(\xi-z)^{n+1}}\,d\xi,

where

C

is a simple closed contour having positive orientation and

z

lies inside

C