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

[uaefame]

Homework Statement

The bessel function J_n(x) is defined by the integral
J_n(x)=1/(iⁿπ)∫₀^πe^ixcosφcos(nφ)dφ

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

Homework Equations

The Attempt at a Solution

I tried combining the cos and exp term together
J_n(x) = ∫_-π^π e^{i(xcosφ+nφ)}dφ

 

[Mark44]

Homework Statement

The bessel function J_n(x) is defined by the integral
J_n(x)=1/(iⁿπ)∫₀^πe^ixcosφcos(nφ)dφ

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

Homework Equations

The Attempt at a Solution

I tried combining the cos and exp term together
J_n(x) = ∫_-π^π e^{i(xcosφ+nφ)}dφ

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

 

[Ray Vickson]

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

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.