How would I go about showing the following result to be true using contour integration?

Question

$$ \int_{-\infty}^{\infty} dk e^{-ak^2+bk} = \sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}} $$

As always any help appreciated - thank you!

Answer 1

Use the integral of the exponential function, the function of $\exp$ is $$ e^{-ak^2 - bk} = e^{-(ak^2 - 2bk\frac{1}{2}\frac{\sqrt{a}}{\sqrt{a}}+b^2\frac{1}{4}\frac{1}{a})} \cdot e^{b^2\frac{1}{4}\frac{1}{a}} = e^{-(\sqrt{a}k -\frac{b}{2 \sqrt{a}})^2} \cdot e^{b^2\frac{1}{4}\frac{1}{a}} $$