Gaussian integral with complex linear component

Question

I want to prove this gaussian integral is equivalent to left side using cauchy's theorem: $$I=\int^\infty_{-\infty} dx \exp(-ax^2+bx)=\sqrt{\frac{\pi}{a}}e^{b^2/4a} $$ with $a \in\mathbb R$, $a>0$ and $b=\beta + i \nu$.

Answer 1

$$\int^\infty_{-\infty}e^{-ax^2+bx} =\int^\infty_{-\infty}e^{-(\sqrt ax-\frac{b}{2\sqrt a})^2-\frac{b^2}{4a}}$$ Using the substitution $u=\sqrt ax-\frac{b}{2\sqrt a}$,$du=adx$ $$\frac{e^{-\frac{b^2}{2a}}}{\sqrt a}\int^\infty_{-\infty}e^{-u^2}du=\sqrt{\frac{\pi}{a}}e^{-\frac{b^2}{2a}}$$