Error function
Derivative and integral

The derivative of the error function follows immediately from its definition: d d z erf z = 2 π e − z 2. {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}.} From this, the derivative of the imaginary error function is also immediate: d d z erfi z = 2 π e z 2. {\displaystyle {\frac {d}{dz}}\operatorname {erfi} z={\frac {2}{\sqrt {\pi }}}e^{z^{2}}.} An antiderivative of the error function, obtainable by integration by parts, is z erf z + e − z 2 π. {\displaystyle z\operatorname {erf} z+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}.} An antiderivative of the imaginary error function, also obtainable by integration by parts, is z erfi z − e z 2 π. {\displaystyle z\operatorname {erfi} z-{\frac {e^{z^{2}}}{\sqrt {\pi }}}.} Higher order derivatives are given by erf (k) z = 2 (− 1) k − 1 π H k − 1 (z) e − z 2 = 2 π d k − 1 d z k − 1 (e − z 2), k = 1, 2, … {\displaystyle \operatorname {erf} ^{(k)}z={\frac {2(-1)^{k-1}}{\sqrt {\pi }}}{\mathit {H}}_{k-1}(z)e^{-z^{2}}={\frac {2}{\sqrt {\pi }}}{\frac {\mathrm {d} ^{k-1}}{\mathrm {d} z^{k-1}}}\left(e^{-z^{2}}\right),\qquad k=1,2,\dots } where H are the physicists' Hermite polynomials.
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