I want to solve the following integral:
$$ f_{X}(x) = \int_{-\infty}^{\infty} \exp\left( \frac{-j (N-1) \alpha t - (N+1) \alpha \beta t^{2}} {(1- j \beta t) (1+j \beta t)} \right) \frac{e^{jxt}}{(1-j\beta t)^{N}(1+j \beta t)^{N}} dt $$
where $\alpha , \beta \in \mathbb{R}_{+}$ and $N \in \mathbb{Z}_{++}$ and $j = \sqrt{-1}$.
However, I am not sure how to deal with the $t^{2}$ in the exponential.
It seems to suggest completing the square, but perhaps I am failing in my algebra, because I can't get anywhere that way.
The completing the square idea:
We expand the integrand as follows:
$$ \exp\left(\frac{-j(N-1) \alpha t}{(1- j \beta t) (1 + j \beta t)}\right) \exp\left(\frac{-(N+1) \alpha \beta t^{2}}{(1- j \beta t) (1 + j \beta t)}\right) \exp\left(\frac{jxt(1- j \beta t) (1 + j \beta t)}{(1- j \beta t) (1 + j \beta t)}\right)$$
Then I try to eliminate all the imaginary numbers except for the coefficients of $xt$, so we can take a Fourier transform of a rational function and use the resideu thm. However, this is where I get stuck....perhaps this is not the right approach.
Anyway...Mathematica seems to have failed to find an answer.
Any ideas?