I want a closed form/ semi closed form of the expected value of a complicated function.
The function looks like this,
$$ f(x) = \frac{\sin(A \frac{x}{2})}{\sin(\frac{x}{2})} \frac{\sin(M B\frac{x}{2})}{\sin(B \frac{x}{2})} \cos\Big[\frac{x}{2}[(A - 1) + B(M - 1)]\Big] $$
The terms $A$, $B$, $M$ are constants and integers.
Where, $x$ comes from a Gaussian distribution.
$$ x \sim \mathcal{N}(\mu, \sigma) $$
I want to know what is E[f(x)]. How to approach it?
In integral form, I want to solve the following integral.
$$ E[f(x)] = \int_{-\pi}^{+\pi} \frac{\sin(A \frac{x}{2})}{\sin(\frac{x}{2})} \frac{\sin(M B\frac{x}{2})}{\sin(B \frac{x}{2})} \cos\Big[\frac{x}{2}[(A - 1) + B(M - 1)]\Big] \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{(2\sigma^2)}} dx $$
Mathematica is not able to compute this. Can this integral be performed in the Complex domain?