Asymptotic expansion of Legendre polynomial

Question

I am trying to determine the asymptotic expansion of the Legendre function as described in a Bender and Orszag text, but have been unable to - some solutions are online, but they approach it in different forms and I'd like to solve it using the below integral definition of the Legendre polynomial.

Specifically, defining:

$$P_n(x)=\frac{1}{\pi}\int_0^\infty(x+cos(\theta)\sqrt{x^2-1})^n d\theta $$

Show that for large n:

$$P_n(x) \sim \frac{1}{\sqrt{2\pi n}}\frac{(x+(x^2-1)^{1/2})^{n+1/2}}{(x^2-1)^{1/4}}$$

Any help is appreciated!

Answer 1

This is an application of the famous saddle-point approximation. Write $$ P_n(x)=\int e^{nf(\theta)}d\theta$$ with $f(\theta)=\log(x+\cos(\theta)\sqrt{x^2-1})$.

Saddle-point gives $$P_n(x)\approx\sqrt{\frac{2\pi}{nf''(\theta_0)}}e^{nf(\theta_0)},$$ where $\theta_0$ is the solution to $f'(\theta_0)=0$.

In your case $\theta_0=0$ and this formula gives what you are looking for.