---
tag: XE3913G5SA
kind: definition
name: Faddeeva function
msc: ["33B20"]
defines: 3GHGFTMFQR
uses: [TZ3CT1YA9B, 0G4F0XB9X2]
sources:
  - url: https://en.wikipedia.org/wiki/Faddeeva_function
---

## Definition

Let $\operatorname{erfc}$ be the [[complementary error function|TZ3CT1YA9B]], extended to complex arguments as an [[entire function|0G4F0XB9X2]].
The *Faddeeva function* is the entire function $w \colon \C \to \C$ given by
$$w(z) = e^{-z^2} \operatorname{erfc}(-iz).$$

## Remarks

For $\operatorname{Im} z > 0$ it has the integral form
$$w(z) = \frac{i}{\pi} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{z - t}\, dt.$$

Example: $w(0) = \operatorname{erfc} 0 = 1$.
For real $x$, $\operatorname{erfc}(-ix) = 1 + i \operatorname{erfi} x$ with $\operatorname{erfi} x = \frac{2}{\sqrt{\pi}}\int_0^x e^{t^2}\, dt$, so
$$w(x) = e^{-x^2} + \frac{2i}{\sqrt{\pi}} F(x),$$
where $F$ is the [[Dawson function|EQ4QB1QW60]].

Non-example: $z \mapsto e^{-z^2} \operatorname{erfc}(z)$ is a different function, because the argument of $\operatorname{erfc}$ is not rotated.
At $z = i$ it equals $e \operatorname{erfc}(i) = e\,(1 - i \operatorname{erfi} 1)$, which is not real, while $w(i) = e \operatorname{erfc} 1$ is real.

The real part of $w(x + iy)$ for $y > 0$ is, up to a constant factor, the Voigt profile, the convolution of a Gaussian with a Lorentzian.