I want to compute the integral $$\int_\mathbb{R}e^{-t\left(y-\dfrac{(at+x)i}{2t}\right)^2}dy$$
I know that $\int_\mathbb{R}e^{-ty^2}dy=\sqrt{\pi/t}$, but here there is an extra imaginary factor. What can I do?
2026-05-15 20:48:16.1778878096
Integral of exponential with second degree exponent
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Integrate around a contour. Start at $-R$. Go in a straight line to $R$. Then go to $R + bi$ (let me use $b$ for the imaginary bit), then go to $-R+bi$, then finally go back to $-R$. Show that the integral of the bits from $R$ to $R+bi$ and $-R+bi$ to $-R$ converge to zero as $R \to +\infty$. You know the integral around the contour is zero. The integral along one of the horizontal lines is the integral you want, and the integral along the other horizontal line is the integral you know how to do. And so they must equal each other.