Evaluating $\int_{-\pi}^{\pi} dq \; e^{iqn} e^{i \alpha\cos(q)}$

Question

I am trying to solve this definite integral

$$\int_{-\pi}^{\pi} dq \; e^{iqn} e^{i \alpha\cos(q)}$$

where $\alpha \in\mathbb{R} $ and $n\in\mathbb{Z}$.

I know that for $n=0$

$$\int_{-\pi}^{\pi} dq \; e^{i \alpha\,cos(q)} = 2\pi J_0(|\alpha|)$$

where $J_0$ is the Bessel function of the first kind. I tried to use the knowledge of the second integral to solve the first one (e.g., integration by parts) but without success so far.

Any suggestion?

Answer 1

Hint

You need to remember that , for integer values of $n$, another integral representation of the Bessel function $J_n(\alpha)$ is $$J_n(\alpha) = \frac{1}{2\pi} \int_{-\pi}^{+\pi} e^{i(\alpha \sin(q) -n q)} \,dq=\frac{1}{2\pi} \int_{-\pi}^{+\pi} e^{-in q}\,e^{i\alpha \sin(q) }\,dq$$ It even seems that this was the form that Bessel used (have a look here).