Error function
Imaginary error function

The imaginary error function, denoted erfi, is defined as

erfi x = − i erf i x = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D (x), {\displaystyle {\begin{aligned}\operatorname {erfi} x&=-i\operatorname {erf} ix\\&={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{t^{2}}\,\mathrm {d} t\\&={\frac {2}{\sqrt {\pi }}}e^{x^{2}}D(x),\end{aligned}}} where D (x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow).

Despite the name "imaginary error function", erfi x is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w (z) = e − z 2 erfc (− i z) = erfcx (− i z). {\displaystyle w(z)=e^{-z^{2}}\operatorname {erfc} (-iz)=\operatorname {erfcx} (-iz).}
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