# How to solve integral $\int_0^{2\pi} e^{i(a\cos\phi + b\sin\phi)} \cos\phi\ d\phi$

I'm trying to solve $\int_0^{2\pi} e^{i(a\cos\phi + b\sin\phi)} \cos\phi\ d\phi$ for a radiation problem in physics. In the special case that $b = 0$, this reduces to a Bessel function of the first kind, but I have no clue where to start for this. Any ideas?

• Idea: $a \cos \phi+b \sin \phi = \sqrt{a^2+b^2} \cos(\phi-\phi_0)$ and then translate $\phi$. But you'll get a factor $\cos(\phi+\phi_0)$ outside which you would possibly have to develop. – H. H. Rugh Aug 1 '16 at 22:08
• Bessel: $\displaystyle{\,\mathrm{J}_{1}}$. – Felix Marin Aug 1 '16 at 22:19
• Split the exponential into two exponentials. Series expand the exponentials, most terms will vanish. – R. Rankin Aug 1 '16 at 23:25
• @FelixMarin. Bessel $2 i \pi J_1(a)$ if $b=0$ I suppose. $b$ seems to be very problematic (at least to me). – Claude Leibovici Aug 2 '16 at 2:24