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  "body": "\n## Definition\n\nLet $\\operatorname{erfc}$ be the [[complementary error function|TZ3CT1YA9B]], extended to complex arguments as an [[entire function|0G4F0XB9X2]].\nThe *Faddeeva function* is the entire function $w \\colon \\C \\to \\C$ given by\n$$w(z) = e^{-z^2} \\operatorname{erfc}(-iz).$$\n\n## Remarks\n\nFor $\\operatorname{Im} z \u003e 0$ it has the integral form\n$$w(z) = \\frac{i}{\\pi} \\int_{-\\infty}^{\\infty} \\frac{e^{-t^2}}{z - t}\\, dt.$$\n\nExample: $w(0) = \\operatorname{erfc} 0 = 1$.\nFor 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\n$$w(x) = e^{-x^2} + \\frac{2i}{\\sqrt{\\pi}} F(x),$$\nwhere $F$ is the [[Dawson function|EQ4QB1QW60]].\n\nNon-example: $z \\mapsto e^{-z^2} \\operatorname{erfc}(z)$ is a different function, because the argument of $\\operatorname{erfc}$ is not rotated.\nAt $z = i$ 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.\n\nThe real part of $w(x + iy)$ for $y \u003e 0$ is, up to a constant factor, the Voigt profile, the convolution of a Gaussian with a Lorentzian.\n",
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