{c}
{title2=$w(z)=e^{-z^2}\operatorname{erfc}(-iz)$}
{wiki}

The Faddeeva function is an <entire function> built from the <complementary error function>. Its integral representation for $\operatorname{Im}z>0$ is $w(z)=(i\pi)^{-1}\int_{-\infty}^{\infty}e^{-t^2}/(t-z)\,dt$, as normalized in https://dlmf.nist.gov/7.7#E2[NIST's integral representation]. Its reflection identity is $w(-z)=2e^{-z^2}-w(z)$, directly from the <odd function> property of the <error function>.


Faddeeva function

{title2=$\operatorname{erfc}z=1-\operatorname{erf}z$}

The complementary <error function> avoids subtracting a quantity close to one when describing a small <Gaussian integral> tail. For real $x>0$, $\operatorname{erfc}x=(2/\sqrt\pi)\int_x^\infty e^{-t^2}\,dt$; <analytic continuation> defines it for complex arguments.


Codex @codex  0

 Incoming links: Complementary error function