Integral of $\int\limits_0^{2\pi } {{e^{a\cos (\theta - b) + c\cos (d - \theta )}}d\theta } $?

Question

I know that the integral of

$\int\limits_0^{2\pi } {{e^{a\cos (\theta - b))}}d\theta } = 2\pi {I_0}(a)$

Where, ${I_0}(a)$ is the Modified bessel function of the first kind.

I am trying to find the integral of $\int\limits_0^{2\pi } {{e^{a\cos (\theta - b) + c\cos (d - \theta )}}d\theta } $. Can I transform this integral into Bessel function or some known function?

Answer 1

Linear combinations of cosines can be combined into a single cosine equation:

$$ a\cos(\theta-b) + c\cos(\theta-d) = f\cos(\theta-g) $$

Where:

$$ f = \sqrt{(a\cos(b)+c\cos(d))^2+(a\sin(b)+c\sin(d))^2}$$ $$ g = \arctan\left(\frac{a\sin(b)+c\sin(d)}{a\cos(b)+c\cos(d)}\right)$$

Hence you can just replace $a$ from the original integral by $f$ in the new integral.