Editing Error function (section)

==Related functions==

===Complementary error function===
The '''complementary error function''', denoted {{math|erfc}}, is defined as
[[File:Plot of the complementary error function Erfc(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the complementary error function Erfc(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the complementary error function Erfc(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
<math display="block">\begin{align}
\operatorname{erfc} x
& = 1-\operatorname{erf} x \\[5pt]
& = \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\,\mathrm  dt \\[5pt]
& = e^{-x^2} \operatorname{erfcx} x,
\end{align} </math>
which also defines {{math|erfcx}}, the '''scaled complementary error function'''<ref name=Cody93>{{Citation |first=W. J. |last=Cody |title=Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers |url=http://www.stat.wisc.edu/courses/st771-newton/papers/p22-cody.pdf |journal=[[ACM Trans. Math. Softw.]] |volume=19 |issue=1 |pages=22–32 |date=March 1993 |doi=10.1145/151271.151273|citeseerx=10.1.1.643.4394 |s2cid=5621105 }}</ref> (which can be used instead of {{math|erfc}} to avoid [[arithmetic underflow]]<ref name=Cody93/><ref name=Zaghloul07>{{Citation |first=M. R. |last=Zaghloul |title=On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand | journal = [[Monthly Notices of the Royal Astronomical Society]] |volume=375 |issue=3 |pages=1043–1048 |date=1 March 2007 |doi=10.1111/j.1365-2966.2006.11377.x|bibcode=2007MNRAS.375.1043Z |doi-access=free }}</ref>). Another form of {{math|erfc ''x''}} for {{math|''x'' ≥ 0}} is known as Craig's formula, after its discoverer:<ref>John W. Craig, [http://wsl.stanford.edu/~ee359/craig.pdf ''A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations''] {{Webarchive|url=https://web.archive.org/web/20120403231129/http://wsl.stanford.edu/~ee359/craig.pdf |date=3 April 2012 }}, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.</ref>
<math display="block">\operatorname{erfc} (x \mid x\ge 0)
= \frac{2}{\pi} \int_0^\frac{\pi}{2} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) \, \mathrm d\theta.</math>
This expression is valid only for positive values of {{mvar|x}}, but it can be used in conjunction with {{math|erfc ''x'' {{=}} 2 − erfc(−''x'')}} to obtain {{math|erfc(''x'')}} for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the {{math|erfc}} of the sum of two non-negative variables is as follows:<ref>{{cite journal |doi=10.1109/TCOMM.2020.2986209 |title=A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis|journal=IEEE Transactions on Communications |volume=68 |issue=7 |pages=4117–4125 |year=2020 |last1=Behnad |first1=Aydin |s2cid=216500014}}</ref>
<math display="block">\operatorname{erfc} (x+y \mid x,y\ge 0) = \frac{2}{\pi} \int_0^\frac{\pi}{2} \exp \left( - \frac{x^2}{\sin^2 \theta} - \frac{y^2}{\cos^2 \theta} \right) \,\mathrm  d\theta.</math>

===Imaginary error function===
The '''imaginary error function''', denoted {{math|erfi}}, is defined as
[[File:Plot of the imaginary error function Erfi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the imaginary error function Erfi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the imaginary error function Erfi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
<math display="block">\begin{align}
\operatorname{erfi} x 
& = -i\operatorname{erf} ix \\[5pt]
& = \frac{2}{\sqrt\pi} \int_0^x e^{t^2}\,\mathrm dt \\[5pt]
& = \frac{2}{\sqrt\pi} e^{x^2} D(x),
\end{align} </math>
where {{math|''D''(''x'')}} is the [[Dawson function]] (which can be used instead of {{math|erfi}} to avoid [[arithmetic overflow]]<ref name=Cody93/>).

Despite the name "imaginary error function", {{math|erfi ''x''}} is real when {{mvar|x}} is real.

When the error function is evaluated for arbitrary [[complex number|complex]] arguments {{mvar|z}}, the resulting '''complex error function''' is usually discussed in scaled form as the [[Faddeeva function]]:
<math display="block">w(z) = e^{-z^2}\operatorname{erfc}(-iz) = \operatorname{erfcx}(-iz).</math>

===Cumulative distribution function===
The error function is essentially identical to the standard [[normal cumulative distribution function]], denoted {{math|Φ}}, also named {{math|norm(''x'')}} by some software languages{{Citation needed|date=July 2020}}, as they differ only by scaling and translation. Indeed,
[[File:Normal cumulative distribution function complex plot in Mathematica 13.1 with ComplexPlot3D.svg|alt=the normal cumulative distribution function plotted in the complex plane|thumb|the normal cumulative distribution function plotted in the complex plane]]
<math display="block">\begin{align}
\Phi(x) 
&= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^\tfrac{-t^2}{2}\,\mathrm dt\\[6pt] 
&= \frac{1}{2} \left(1+\operatorname{erf}\frac{x}{\sqrt 2}\right)\\[6pt]
&= \frac{1}{2} \operatorname{erfc}\left(-\frac{x}{\sqrt 2}\right)
\end{align}</math>
or rearranged for {{math|erf}} and {{math|erfc}}:
<math display="block">\begin{align}
  \operatorname{erf}(x)  &= 2 \Phi{\left ( x \sqrt{2} \right )} - 1 \\[6pt]
  \operatorname{erfc}(x) &= 2 \Phi{\left ( - x \sqrt{2} \right )} \\
 &= 2\left(1 - \Phi{\left ( x \sqrt{2} \right)}\right).
\end{align}</math>

Consequently, the error function is also closely related to the [[Q-function]], which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as
<math display="block">\begin{align}
Q(x) &= \frac{1}{2} - \frac{1}{2} \operatorname{erf} \frac{x}{\sqrt 2}\\
&= \frac{1}{2}\operatorname{erfc}\frac{x}{\sqrt 2}.
\end{align}</math>

The [[inverse function|inverse]] of {{math|Φ}} is known as the [[Quantile function|normal quantile function]], or [[probit]] function and may be expressed in terms of the inverse error function as
<math display="block">\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\operatorname{erfc}^{-1}(2p).</math>

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

The error function is a special case of the [[Mittag-Leffler function]], and can also be expressed as a [[confluent hypergeometric function]] (Kummer's function):
<math display="block">\operatorname{erf} x = \frac{2x}{\sqrt\pi} M\left(\tfrac{1}{2},\tfrac{3}{2},-x^2\right).</math>

It has a simple expression in terms of the [[Fresnel integral]].{{Elucidate|date=May 2012}}

In terms of the [[regularized gamma function]] {{mvar|P}} and the [[incomplete gamma function]],
<math display="block">\operatorname{erf} x
= \sgn x \cdot P\left(\tfrac{1}{2}, x^2\right)
= \frac{\sgn x}{\sqrt\pi} \gamma{\left(\tfrac{1}{2}, x^2\right)}.</math>{{math|sgn ''x''}} is the [[sign function]].
===Iterated integrals of the complementary error function===
The iterated integrals of the complementary error function are defined by<ref>{{cite book | last1 = Carslaw | first1 = H. S. |author1-link = Horatio Scott Carslaw | last2 = Jaeger | first2 = J. C.| author2-link = John Conrad Jaeger | year = 1959 | title = Conduction of Heat in Solids | edition = 2nd | publisher = Oxford University Press | isbn = 978-0-19-853368-9 | page = 484}}</ref>
<math display="block">\begin{align}
i^n\!\operatorname{erfc} z &= \int_z^\infty  i^{n-1}\!\operatorname{erfc} \zeta\,\mathrm  d\zeta \\[6pt]
i^0\!\operatorname{erfc} z &= \operatorname{erfc} z \\
i^1\!\operatorname{erfc} z &= \operatorname{ierfc} z = \frac{1}{\sqrt\pi} e^{-z^2} - z \operatorname{erfc} z \\
i^2\!\operatorname{erfc} z &= \tfrac{1}{4} \left( \operatorname{erfc} z -2 z \operatorname{ierfc} z \right) \\
\end{align}</math>

The general recurrence formula is
<math display="block">2 n \cdot i^n\!\operatorname{erfc} z = i^{n-2}\!\operatorname{erfc} z -2 z \cdot i^{n-1}\!\operatorname{erfc} z</math>

They have the power series
<math display="block">i^n\!\operatorname{erfc} z =\sum_{j=0}^\infty \frac{(-z)^j}{2^{n-j}j! \,\Gamma \left( 1 + \frac{n-j}{2}\right)},</math>
from which follow the symmetry properties
<math display="block">i^{2m}\!\operatorname{erfc} (-z) =-i^{2m}\!\operatorname{erfc} z +\sum_{q=0}^m \frac{z^{2q}}{2^{2(m-q)-1}(2q)! (m-q)!}</math>
and
<math display="block">i^{2m+1}\!\operatorname{erfc}(-z) =i^{2m+1}\!\operatorname{erfc} z +\sum_{q=0}^m \frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)! (m-q)!}.
</math>