A quick question about Gaussians.
Normally Gaussians with quadratic terms are defined as
$\int dx e^{-ax^{2} + bx} = \sqrt{\frac{\pi}{a}}e^{\frac{b^{2}}{4a}}$.
However, what happens if $b$ can be complex valued? More precisely, can the equation be modified as $\int dx e^{-ax^{2} + bx} = \sqrt{\frac{\pi}{a}}e^{\frac{\bar{b}b}{4a}}$?
Kind regards,
Livius
Where does the $\frac {b^2}{4a}$ factor come from?
$-ax^2 + bx = -a(x-\frac {b}{2a})^2 + \frac {b^2}{4a}$
Does a complex $b$ change the numerator of the $\frac {b^2}{4a}$ term to $\frac {|b|^2}{4a}$ ?