So I am complicating some coherence problem and I wish to integrate the overlap of two wave functions with some phase induced.
The result is: enter image description here
In the second part, the cos replaced the complex exponential where its odd sin component cancels out over the symmetric integral. Also, a is negative and b is general.
Does anyone know what the result of this integral is? Also, I would reaaaly love to know where and/or how to be able to get the result, as I want to complicate my life even more at later times.
The point is that, originally, the integral had b=0, so it was a Gaussian with a cosine, but I do not know how to calculate this.
I'm not entirely clear on which integral you want to do, but I gather you're interested in integrals of the form
$$\int_{-\infty}^\infty d x \exp(a x^2 + b x + c)$$
where $a$ is a negative real parameter and $b,c$ are complex numbers.
The trick is to use a change of variables and complete the square, so that you have an integral of the form
$$D \int_{-\infty}^\infty d x \exp(a (x-x_0)^2)$$
and then do another change of variables to eliminate $x_0$ and use standard Gaussian integral techniques such as looking it up or transforming to a 2D integral with polar coordinates to finish the problem.