Error function
Inverse functions

Given a complex number z, there is not a unique complex number w satisfying erf w = z, so a true inverse function would be multivalued. However, for −1 < x < 1, there is a unique real number denoted erf x satisfying erf (erf − 1 x) = x. {\displaystyle \operatorname {erf} \left(\operatorname {erf} ^{-1}x\right)=x.}

The inverse error function is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk | z | < 1 of the complex plane, using the Maclaurin series erf − 1 z = ∑ k = 0 ∞ c k 2 k + 1 (π 2 z) 2 k + 1, {\displaystyle \operatorname {erf} ^{-1}z=\sum _{k=0}^{\infty }{\frac {c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},} where c = 1 and c k = ∑ m = 0 k − 1 c m c k − 1 − m (m + 1) (2 m + 1) = { 1, 1, 7 6, 127 90, 4369 2520, 34807 16200, … }. {\displaystyle {\begin{aligned}c_{k}&=\sum _{m=0}^{k-1}{\frac {c_{m}c_{k-1-m}}{(m+1)(2m+1)}}\\&=\left\{1,1,{\frac {7}{6}},{\frac {127}{90}},{\frac {4369}{2520}},{\frac {34807}{16200}},\ldots \right\}.\end{aligned}}}

So we have the series expansion (common factors have been canceled from numerators and denominators): erf − 1 z = π 2 (z + π 12 z 3 + 7 π 2 480 z 5 + 127 π 3 40320 z 7 + 4369 π 4 5806080 z 9 + 34807 π 5 182476800 z 11 + ⋯). {\displaystyle \operatorname {erf} ^{-1}z={\frac {\sqrt {\pi }}{2}}\left(z+{\frac {\pi }{12}}z^{3}+{\frac {7\pi ^{2}}{480}}z^{5}+{\frac {127\pi ^{3}}{40320}}z^{7}+{\frac {4369\pi ^{4}}{5806080}}z^{9}+{\frac {34807\pi ^{5}}{182476800}}z^{11}+\cdots \right).} (After cancellation the numerator and denominator values in OEIS: A092676 and OEIS: A092677 respectively; without cancellation the numerator terms are values in OEIS: A002067.) The error function's value at ±∞ is equal to ±1.

For | z | < 1, we have erf(erf z) = z.

The inverse complementary error function is defined as erfc − 1 (1 − z) = erf − 1 z. {\displaystyle \operatorname {erfc} ^{-1}(1-z)=\operatorname {erf} ^{-1}z.} For real x, there is a unique real number erfi x satisfying erfi(erfi x) = x. The inverse imaginary error function is defined as erfi x.

For any real x, Newton's method can be used to compute erfi x, and for −1 ≤ x ≤ 1, the following Maclaurin series converges: erfi − 1 z = ∑ k = 0 ∞ (− 1) k c k 2 k + 1 (π 2 z) 2 k + 1, {\displaystyle \operatorname {erfi} ^{-1}z=\sum _{k=0}^{\infty }{\frac {(-1)^{k}c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},} where c is defined as above.
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