# I Gaussian integration for complex phase

1. Oct 17, 2016

### spaghetti3451

I would like to prove that

$\displaystyle{\int dx'\ \frac{1}{\sqrt{AB}}\exp\bigg[i\frac{(x''-x')^{2}}{A}\bigg]\exp\bigg[i\frac{(x'-x)^{2}}{B}\bigg]=\frac{1}{\sqrt{A+B}}\exp\bigg[i\frac{(x''-x)^{2}}{A+B}\bigg]}$

Is there an easy way to do this integration that does not involve squaring the brackets?

2. Oct 18, 2016

### stevendaryl

Staff Emeritus
Hmm. Completing the square would seem to me by far to be the easiest way to do it. Why do you want a different way?

A second comment: Are you sure that is correct? Gaussian integrals usually involve factors of $\sqrt{2\pi}$