Is there a way to get a hint about these kind of integrations?
$$\int_0^{2\pi}e^{ik\cos\theta}d\theta$$ or $$\int_0^{2\pi}e^{k\cos\theta}d\theta$$
Thanks in advance
Edit : I tried that. A simple change of variable $\cos{\theta}= x$: $$\int_0^{\pi}e^{k\cos\theta}d\theta=\int_{-1}^{1}\frac{e^{kx}}{\sqrt{1-x^2}}dx=\int_{-1}^{1}\arcsin'(x)e^{kx}dx \\ =[\arcsin(x)e^{kx}]^{1}_{-1}-k\int_{-1}^{1}\arcsin(x)e^{kx}dx \\ =[\arcsin(x)e^{kx}]^{1}_{-1}-[k(\sqrt{1-x^2}+x\arcsin(x))e^{kx}]^{1}_{-1}+k^2 \int_{-1}^{1}(\sqrt{1-x^2}+x\arcsin(x))e^{kx}dx$$
It looks right but it doesn't look like it's leading anywhere