Computing an integral related to the J-Bessel function

Question

It is known that for real $a,b$, one has the integral representation $$ J_0\left(x\sqrt{a^2+b^2}\right) = \frac{1}{2\pi} \int_{0}^{2\pi} e^{i x(a \cos(t) +b\sin (t))} dt, $$ where $J_0$ is the usual $J$-Bessel function of order $0$. I'm interested in the slightly modified integral $$ I_n(x) := \frac{1}{2\pi} \int_{0}^{2\pi} e^{i n t} e^{i x(a \cos(t) +b\sin (t))} dt, $$ where $n$ is (in my case) an integer. Given my limited knowledge of Bessel functions, this certainly seems like it should be related to higher order $J$-Bessel functions. However, I have not found any references for this specific integral, and Mathematica cannot evaluate the integral either. Has anyone seen this integral before, and/or can someone explain how to relate this to other Bessel functions? Many thanks in advance.

Answer 1

Let $r=\sqrt{a^2+b^2}$, $\sin\phi=a/r$, $\cos\phi=b/r$. Then, substituting $t+\phi=\tau$, $$I_n(x)=\frac1{2\pi}\int_0^{2\pi}e^{int+irx\sin(t+\phi)}\,dt=\frac{e^{-in\phi}}{2\pi}\int_0^{2\pi}e^{in\tau+irx\sin\tau}\,d\tau=e^{in(\pi-\phi)}J_n(rx)$$ (with the limits of integration unchanged, since we're integrating over a period).