Integral of exponential of complex expression $\int_{-\infty}^{\infty}\exp\left(-a\left[\left(y+ib/2a\right)^2-i^2b^2/4a^2\right]\right)dy$

Question

I have the following expression

$$ R\int_{-\infty}^{\infty}exp \bigg(-a\bigg[\bigg(y+ib/2a\bigg)^2-i^2b^2/4a^2\bigg]\bigg)dy $$ Where $R$ is real numbers and $i$ denotes complex numbers.

Which should result in the following $$=exp\bigg(-b^2/4a\bigg)\sqrt{\pi/a}$$

I am not sure how to get to that result however, any help would be highly appreciated :)

Answer 1

Hint: $$\int_{-\infty}^{\infty}\exp \bigg(-a\bigg[\bigg(y+ib/2a\bigg)^2-i^2b^2/4a^2\bigg]\bigg)dy=\exp\bigg(-b^2/4a\bigg)\ \int_{-\infty}^{\infty}\exp \bigg(-a\bigg[\bigg(y+ib/2a\bigg)^2\bigg]\bigg)dy$$ now let $\sqrt{a}\bigg(y+ib/2a\bigg)=u$ and after substitution use gamma function. Note that $e^{-ay^2}$ is even.