$$\int_{0}^{2\pi}\exp\left(-iA(x\cos\varphi+y\sin\varphi)\right)\cos(l\varphi)\,d\varphi$$
I'm trying to evaluate the integral for an interference problem in Physics. When $y=0$, this reduces to the Bessel Function of the first kind, and when $l=1$, I can differentiate under the integral w.r.t. $x$ and evaluate the integral (which gives a first order Bessel function of the first kind). However, I'm looking for a more general answer, where $l$ is any integer, and $A$ is an arbitrary constant.
Here's a link to a similar question posted 2 years ago. How to solve integral $\int_0^{2\pi} e^{i(a\cos\phi + b\sin\phi)} \cos\phi\ d\phi$
Best Answer
Let $\rho = \sqrt {x^2 + y^2}, \phi = \arctan(x, y)$. Then $$\int_0^{2 \pi} e^{-i A (x \cos t + y \sin t)} \cos l t \,dt = \\ \int_0^{2 \pi} e^{-i A \rho \cos(t - \phi)} \cos l t \,dt = \cos l \phi \int_0^{2 \pi} e^{-i A \rho \cos t} \cos l t \,dt = \\ 2 \pi (-i)^l \cos l \phi \,J_l(A \rho).$$