Regulated complex Fresnel integral (${\int }_{\mathbb{C}}{e}^{(i\lambda -\alpha )|z{|}^2}{d}^{2}z=\pi /(\alpha -i\lambda )$)

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 NIST definitions.