Editing Error function (section)

===Inverse functions===
[[File:Mplwp erf inv.svg|thumb|300px|Inverse error function]]

Given a complex number {{mvar|z}}, there is not a ''unique'' complex number {{mvar|w}} satisfying {{math|1=erf ''w'' = ''z''}}, so a true inverse function would be multivalued. However, for {{math|−1 < ''x'' < 1}}, there is a unique ''real'' number denoted {{math|erf<sup>−1</sup> ''x''}} satisfying
<math display="block">\operatorname{erf}\left(\operatorname{erf}^{-1} x\right) = x.</math>

The '''inverse error function''' is usually defined with domain {{open-open|−1,1}}, and it is restricted to this domain in many computer algebra systems.  However, it can be extended to the disk {{math|{{abs|''z''}} < 1}} of the complex plane, using the Maclaurin series<ref>{{cite arXiv | last1 = Dominici | first1 = Diego | title = Asymptotic analysis of the derivatives of the inverse error function | eprint = math/0607230 | year = 2006}}</ref>
<math display="block">\operatorname{erf}^{-1} z=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt\pi}{2}z\right )^{2k+1},</math>
where {{math|1=''c''<sub>0</sub> = 1}} and
<math display="block">\begin{align}
c_k & =\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} \\[1ex]
&= \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}.
\end{align}</math>

So we have the series expansion (common factors have been canceled from numerators and denominators):
<math display="block">\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 ).</math>
(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&nbsp;{{math|±∞}} is equal to&nbsp;{{math|±1}}.

For {{math|{{abs|''z''}} < 1}}, we have {{math|1=erf(erf<sup>−1</sup> ''z'') = ''z''}}.

The '''inverse complementary error function''' is defined as
<math display="block">\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1} z.</math>
For real {{mvar|x}}, there is a unique ''real'' number {{math|erfi<sup>−1</sup> ''x''}} satisfying {{math|1=erfi(erfi<sup>−1</sup> ''x'') = ''x''}}.  The '''inverse imaginary error function''' is defined as {{math|erfi<sup>−1</sup> ''x''}}.<ref>{{cite arXiv | last1 = Bergsma | first1 = Wicher | title = On a new correlation coefficient, its orthogonal decomposition and associated tests of independence | eprint = math/0604627 | year = 2006}}</ref>

For any real ''x'', [[Newton's method]] can be used to compute {{math|erfi<sup>−1</sup> ''x''}}, and for {{math|−1 ≤ ''x'' ≤ 1}}, the following Maclaurin series converges:
<math display="block">\operatorname{erfi}^{-1} z =\sum_{k=0}^\infty\frac{(-1)^k c_k}{2k+1} \left( \frac{\sqrt\pi}{2} z \right)^{2k+1},</math>
where {{math|''c''<sub>''k''</sub>}} is defined as above.