Complex Analysis: Theory and Applications · Problem 6.2.19

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · ml-inequality

Problem

Use the ML inequality to show that

|P_n(x)| \leq 1

, where

P_n

is the

n^{th}

Legendre polynomial defined on

-1 \leq x \leq 1

P_n(x) = \frac{1}{\pi}\int_0^\pi \left(x + i\sqrt{1-x^2}\cos\theta\right)^n d\theta.