Editing Error function (section)

===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>