Gaussian integration for complex phase

[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?

 

[stevendaryl]

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?

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 $\sqrt{2\pi}$