= Regulated complex Fresnel integral
{title2=$\int_{\mathbb C}e^{(i\lambda-\alpha)|z|^2}d^2z=\pi/(\alpha-i\lambda)$}
For $\alpha>0$, polar coordinates reduce the integral to $2\pi\int_0^\infty r e^{-(\alpha-i\lambda)r^2}dr$. The result is $\pi/(\alpha-i\lambda)$. Its damped limit for real $\lambda>0$ is $i\pi/\lambda$, which fixes the sign of the oscillatory complex <Gaussian integral>. The <Fresnel integral> convention can be compared with https://dlmf.nist.gov/7.2[NIST definitions].
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