Knowing that $$\int_{0}^{2\pi} e^{x\cos \theta } d\theta = 2\pi I_0(x)$$ where $I_0$ is the modified Bessel function. Is there a way/trick to find an analytical expression for $$\int_{0}^{2\pi} e^{x\cos \theta + y\cos \theta} d\theta.$$
2026-03-26 18:50:41.1774551041
What is the integral of $e^{x\cos\theta + y\cos\theta}$
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Why would this not be $$ \int_0^{2\pi} e^{x\cos t + y \cos t} dt = \int_0^{2\pi} e^{(x+y)\cos t} dt = 2\pi I_0(x+y)? $$