Prove approximation given by the physicist Max Born

Question

In an old book about optics, I have found a nice approximation, that for large l one has: $$P_l(\cos(\theta)) \sim \sqrt{\frac{2}{l \pi \sin(\theta)}} \sin \left((l+\frac{1}{2}) \theta + \frac{\pi}{4} \right)$$, where $P_l$ is the l-th Legendre polynomial.

Unfortunately, I do not have any clue how to prove this, but maybe somebody here has an idea and an error approximation for this approximation? A reference would be sufficient too, of course.

Answer 1

You have full asymptotic expansion here, formulas 18.15.12-18.15.13. It can be deduced from the integral representation for Legendre polynomials (formula 18.10.2 here).

However, I think that Born himself deduced this formula by studying the asymptotics of partial wave solutions of the 2D Helmholtz equation.

Answer 2

A reference is e.g. Abramowitz and Stegun, 8.10.7 with $\mu=0, \nu=l,$ and $\cos \left((l+\frac{1}{2}) \theta - \frac{\pi}{4} \right)$ replaced by $\sin \left((l+\frac{1}{2}) \theta + \frac{\pi}{4} \right)$.