Editing Error function (section)

==Properties==
{{multiple image
 | header    =  Plots in the complex plane
 | direction = vertical
 | width     = 250
 | image1    = ComplexExp2.png
 | caption1  = Integrand {{math|exp(−''z''<sup>2</sup>)}}
 | image2    = ComplexErfz.png
 | caption2  = {{math|erf ''z''}}
}}

The property {{math|1=erf (−''z'') = −erf ''z''}} means that the error function is an [[even and odd functions|odd function]]. This directly results from the fact that the integrand {{math|''e''<sup>−''t''<sup>2</sup></sup>}} is an [[even function]] (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).

Since the error function is an [[entire function]] which takes real numbers to real numbers, for any [[complex number]] {{mvar|z}}:
<math display="block">\operatorname{erf} \overline{z} = \overline{\operatorname{erf} z} </math>
where <math>\overline{z}
</math> denotes the [[complex conjugate]] of <math>z</math>.

The integrand {{math|1=''f'' = exp(−''z''<sup>2</sup>)}} and {{math|1=''f'' = erf ''z''}} are shown in the complex {{mvar|z}}-plane in the figures at right with [[domain coloring]].

The error function at {{math|+∞}} is exactly 1 (see [[Gaussian integral]]). At the real axis, {{math|erf ''z''}} approaches unity at {{math|''z'' → +∞}} and −1 at {{math|''z'' → −∞}}. At the imaginary axis, it tends to {{math|±''i''∞}}.
<!-- ; the relation {{math|1=erf(−''z'') = −erf ''z''}} holds.!-->

===Taylor series===
The error function is an [[entire function]]; it has no singularities (except that at infinity) and its [[Taylor expansion]] always converges. For {{math|''x'' >> 1}}, however, cancellation of leading terms makes the Taylor expansion unpractical.

The defining integral cannot be evaluated in [[Closed-form expression|closed form]] in terms of [[Elementary function (differential algebra)|elementary functions]] (see [[Liouville's theorem (differential algebra)|Liouville's theorem]]), but by expanding the [[integrand]] {{math|''e''<sup>−''z''<sup>2</sup></sup>}} into its [[Maclaurin series]] and integrating term by term, one obtains the error function's Maclaurin series as:
<math display="block">\begin{align}
\operatorname{erf} z
&= \frac{2}{\sqrt\pi}\sum_{n=0}^\infty\frac{(-1)^n z^{2n+1}}{n! (2n+1)} \\[6pt]
&= \frac{2}{\sqrt\pi} \left(z-\frac{z^3}{3}+\frac{z^5}{10}-\frac{z^7}{42}+\frac{z^9}{216}-\cdots\right)
\end{align}</math>
which holds for every [[complex number]]&nbsp;{{mvar|z}}. The denominator terms are sequence [[oeis:A007680|A007680]] in the [[OEIS]].

For iterative calculation of the above series, the following alternative formulation may be useful:
<math display="block">\begin{align}
\operatorname{erf} z
&= \frac{2}{\sqrt\pi}\sum_{n=0}^\infty\left(z \prod_{k=1}^n {\frac{-(2k-1) z^2}{k (2k+1)}}\right) \\[6pt]
&= \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{z}{2n+1} \prod_{k=1}^n \frac{-z^2}{k}
\end{align}</math>
because {{math|{{sfrac|−(2''k'' − 1)''z''<sup>2</sup>|''k''(2''k'' + 1)}}}} expresses the multiplier to turn the {{mvar|k}}th term into the {{math|(''k''&nbsp;+&nbsp;1)}}th term (considering {{mvar|z}} as the first term).

The imaginary error function has a very similar Maclaurin series, which is:
<math display="block">\begin{align}
\operatorname{erfi} z
 &= \frac{2}{\sqrt\pi}\sum_{n=0}^\infty\frac{z^{2n+1}}{n! (2n+1)} \\[6pt]
 &=\frac{2}{\sqrt\pi} \left(z+\frac{z^3}{3}+\frac{z^5}{10}+\frac{z^7}{42}+\frac{z^9}{216}+\cdots\right)
\end{align}</math>
which holds for every [[complex number]]&nbsp;{{mvar|z}}.

===Derivative and integral===
The derivative of the error function follows immediately from its definition:
<math display="block">\frac{\mathrm d}{\mathrm dz}\operatorname{erf} z =\frac{2}{\sqrt\pi} e^{-z^2}.</math>
From this, the derivative of the imaginary error function is also immediate:
<math display="block">\frac{d}{dz}\operatorname{erfi} z =\frac{2}{\sqrt\pi} e^{z^2}.</math>
An [[antiderivative]] of the error function, obtainable by [[integration by parts]], is
<math display="block">z\operatorname{erf}z + \frac{e^{-z^2}}{\sqrt\pi}+C.</math>
An antiderivative of the imaginary error function, also obtainable by integration by parts, is
<math display="block">z\operatorname{erfi}z - \frac{e^{z^2}}{\sqrt\pi}+C.</math>
Higher order derivatives are given by
<math display="block">\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 dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots</math>
where {{mvar|H}} are the physicists' [[Hermite polynomials]].<ref>{{mathworld|title=Erf|urlname=Erf}}</ref>

===Bürmann series===

An expansion,<ref>{{cite journal|first1=H. M. |last1=Schöpf |first2=P. H. |last2=Supancic |title=On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion |journal=The Mathematica Journal |year=2014 |volume=16 |doi=10.3888/tmj.16-11 |url=http://www.mathematica-journal.com/2014/11/on-burmanns-theorem-and-its-application-to-problems-of-linear-and-nonlinear-heat-transfer-and-diffusion/#more-39602/|doi-access=free }}</ref> which converges more rapidly for all real values of {{mvar|x}} than a Taylor expansion, is obtained by using [[Hans Heinrich Bürmann]]'s theorem:<ref>{{mathworld|urlname=BuermannsTheorem | title = Bürmann's Theorem }}</ref>
<math display="block">\begin{align}
\operatorname{erf} x
&= \frac{2}{\sqrt\pi} \sgn x \cdot \sqrt{1-e^{-x^2}} \left( 1-\frac{1}{12} \left (1-e^{-x^2} \right ) -\frac{7}{480} \left (1-e^{-x^2} \right )^2 -\frac{5}{896} \left (1-e^{-x^2} \right )^3-\frac{787}{276 480} \left (1-e^{-x^2} \right )^4 - \cdots \right) \\[10pt]
&= \frac{2}{\sqrt\pi} \sgn x \cdot \sqrt{1-e^{-x^2}} \left(\frac{\sqrt\pi}{2} + \sum_{k=1}^\infty c_k e^{-kx^2} \right).
\end{align}</math>
where {{math|sgn}} is the [[sign function]]. By keeping only the first two coefficients and choosing {{math|1=''c''<sub>1</sub> = {{sfrac|31|200}}}} and {{math|1=''c''<sub>2</sub> = −{{sfrac|341|8000}}}}, the resulting approximation shows its largest relative error at {{math|1=''x'' = ±1.40587}}, where it is less than 0.0034361:
<math display="block">\operatorname{erf} x \approx \frac{2}{\sqrt\pi}\sgn x \cdot \sqrt{1-e^{-x^2}} \left(\frac{\sqrt{\pi}}{2} + \frac{31}{200}e^{-x^2}-\frac{341}{8000} e^{-2x^2}\right). </math>

===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.

===Asymptotic expansion===
A useful [[asymptotic expansion]] of the complementary error function (and therefore also of the error function) for large real {{mvar|x}} is
<math display="block">\begin{align}
\operatorname{erfc} x &= \frac{e^{-x^2}}{x\sqrt{\pi}}\left(1 + \sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\cdots(2n - 1)}{\left(2x^2\right)^n}\right) \\[6pt] 
&= \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n - 1)!!}{\left(2x^2\right)^n},
\end{align}</math>
where {{math|(2''n'' − 1)!!}} is the [[double factorial]] of {{math|(2''n'' − 1)}}, which is the product of all odd numbers up to {{math|(2''n'' − 1)}}. This series diverges for every finite {{mvar|x}}, and its meaning as asymptotic expansion is that for any integer {{math|''N'' ≥ 1}} one has
<math display="block">\operatorname{erfc} x = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^{N-1} (-1)^n \frac{(2n - 1)!!}{\left(2x^2\right)^n} + R_N(x)</math>
where the remainder is
<math display="block">R_N(x) := \frac{(-1)^N \, (2 N - 1)!!}{\sqrt{\pi} \cdot 2^{N - 1}} \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt,</math>
which follows easily by induction, writing
<math display="block">e^{-t^2} = -\frac{1}{2 t} \, \frac{\mathrm{d}}{\mathrm{d}t} e^{-t^2}</math>
and integrating by parts.

The asymptotic behavior of the remainder term, in [[Landau notation]], is
<math display="block">R_N(x) = O\left(x^{- (1 + 2N)} e^{-x^2}\right)</math>
as {{math|''x'' → ∞}}. This can be found by 
<math display="block">R_N(x) \propto \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt = e^{-x^2} \int_0^\infty (t+x)^{-2N}e^{-t^2-2tx}\,\mathrm dt\leq e^{-x^2} \int_0^\infty x^{-2N} e^{-2tx}\,\mathrm dt \propto x^{-(1+2N)}e^{-x^2}.</math>
For large enough values of {{mvar|x}}, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of {{math|erfc ''x''}} (while for not too large values of {{mvar|x}}, the above Taylor expansion at 0 provides a very fast convergence).

===Continued fraction expansion===
A [[continued fraction]] expansion of the complementary error function was found by [[Pierre-Simon Laplace|Laplace]]:<ref>[[Pierre-Simon Laplace]], [[Traité de mécanique céleste]], tome 4 (1805), livre X, page 255.</ref><ref>{{cite book| last1 = Cuyt | first1 = Annie A. M.|author1-link= Annie Cuyt | last2 = Petersen | first2 = Vigdis B. | last3 = Verdonk | first3 = Brigitte | last4 = Waadeland | first4 = Haakon | last5 = Jones | first5 = William B. | title = Handbook of Continued Fractions for Special Functions | publisher = Springer-Verlag | year = 2008 | isbn = 978-1-4020-6948-2 }}</ref>
<math display="block">\operatorname{erfc} z = \frac{z}{\sqrt\pi}e^{-z^2} \cfrac{1}{z^2+ \cfrac{a_1}{1+\cfrac{a_2}{z^2+ \cfrac{a_3}{1+\dotsb}}}},\qquad a_m = \frac{m}{2}.</math>

===Factorial series===
The inverse [[factorial series]]:
<math display="block">\begin{align}
\operatorname{erfc} z
&= \frac{e^{-z^2}}{\sqrt{\pi}\,z} \sum_{n=0}^\infty \frac{\left(-1\right)^n Q_n}{{\left(z^2+1\right)}^{\bar{n}}} \\[1ex]
&= \frac{e^{-z^2}}{\sqrt{\pi}\,z} \left[1 -\frac{1}{2}\frac{1}{(z^2+1)} + \frac{1}{4}\frac{1}{\left(z^2+1\right) \left(z^2+2\right)} - \cdots \right]
\end{align}</math>
converges for {{math|Re(''z''<sup>2</sup>) > 0}}. Here
<math display="block">\begin{align}
Q_n
&\overset{\text{def}}{{}={}}
\frac{1}{\Gamma{\left(\frac{1}{2}\right)}} \int_0^\infty \tau(\tau-1)\cdots(\tau-n+1)\tau^{-\frac{1}{2}} e^{-\tau} \,d\tau \\[1ex]
&= \sum_{k=0}^n \left(\frac{1}{2}\right)^{\bar{k}} s(n,k),
\end{align}</math>
{{math|''z''<sup>{{overline|''n''}}</sup>}} denotes the [[rising factorial]], and {{math|''s''(''n'',''k'')}} denotes a signed [[Stirling number of the first kind]].<ref>{{cite journal|last=Schlömilch|first=Oskar Xavier | author-link=Oscar Schlömilch|year=1859|title=Ueber facultätenreihen|url=https://archive.org/details/zeitschriftfrma09runggoog | journal=[[:de:Zeitschrift für Mathematik und Physik|Zeitschrift für Mathematik und Physik]] | language=de | volume=4 | pages=390–415}}</ref><ref>{{cite book | last=Nielson | first=Niels | url=https://archive.org/details/handbuchgamma00nielrich | title=Handbuch der Theorie der Gammafunktion | date=1906 | publisher=B. G. Teubner | location=Leipzig|language=de|access-date=2017-12-04|at=p. 283 Eq. 3}}</ref>
There also exists a representation by an infinite sum containing the [[double factorial]]:
<math display="block">\operatorname{erf} z =  \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-2)^n(2n-1)!!}{(2n+1)!}z^{2n+1}</math>