# Find the first 3 terms of the asymptotic expansion of Jn(x)

1. Feb 24, 2016

### uaefame

1. The problem statement, all variables and given/known data
The bessel function Jn(x) is defined by the integral
Jn(x)=1/(inπ)∫0πeixcosφcos(nφ)dφ

From this formula, find the first 3 terms of the asymptotic expansion of Jn(x) when x=n and n is a large positive integer.

2. Relevant equations

3. The attempt at a solution
I tried combining the cos and exp term together
Jn(x) = ∫π ei(xcosφ+nφ)

Last edited by a moderator: Feb 24, 2016
2. Feb 24, 2016

### Staff: Mentor

How is this justified? $e^{\cos(mx)} \cdot \cos(ny) \ne e^{\cos(mx + ny)}$.

3. Feb 24, 2016

### Ray Vickson

Aside from the missing factor outside the integral, and a possible factor of 2 or 1/2, what he wrote is OK because he switched from $\int_{\phi=0}^{\pi} \cdots$ to $\int_{\phi=-\pi}^{\pi} \cdots$, and used the evenness of the $\cos(\phi)$ in the exponential.