Confirmation of definite integral [0, 2 pi] of exp[ r cos(x)+ s sin(x)] cos[a cos(x)+b sin(x)]

Question

I found in a paper the following integral form: $ \int_{0}^{2 \pi} exp[(r \cos(x)+s \sin(x)]\cdot \cos[( a \cos(x)+b \sin(x)] dx $ $= \pi[I_0(\sqrt{C+iD})+I_0(\sqrt{C-iD})]$ with $I_0$ as modified Bessel function, $i$ as imaginary unit and the two factors $ C = r^2+s^2-a^2-b^2$ and $ D = 2 (r a +s b) $. I could not find it in the common tables like Abramovitz or Gradshteyn. Can anyone confirm its truth?

Answer 1

After coming across $\int_{0}^{2 \pi} exp[r \cos(x) + s \sin(x)] dx = 2 \pi I_0(\sqrt{r^2+s^2})$ I found that you just have to split the problem up in the following way:

$I_0(\sqrt{(r+ia)^2+(s+ib)^2}) + I_0(\sqrt{(r-ia)^2+(s-ib)^2}) $

Then considering only the real part and using cos(x) = cos(-x) leads to the given integral representation. The complex parts with $ia$ and $ib$ leads to $ \cos(a \cos(x) + b \sin(x))$ in the integral.