Error function: Difference between revisions

Latest revision as of 13:25, 22 June 2025

Template:Short description Template:Use dmy dates Script error: No such module "Distinguish". In mathematics, the error function (also called the Gauss error function), often denoted by Template:Math, is a function $e r f : ℂ \to ℂ$ defined as:^[1] $\erf (z) = \frac{2}{\sqrt{π}} \int_{0}^{z} e^{- t^{2}} d t .$ Template:Infobox mathematical function

The integral here is a complex contour integral which is path-independent because $\exp (- t^{2})$ is holomorphic on the whole complex plane $ℂ$ . In many applications, the function argument is a real number, in which case the function value is also real.

In some old texts,^[2] the error function is defined without the factor of $\frac{2}{\sqrt{π}}$ . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations.

In statistics, for non-negative real values of Template:Mvar, the error function has the following interpretation: for a real random variable Template:Mvar that is normally distributed with mean 0 and standard deviation $\frac{1}{\sqrt{2}}$ , Template:Math is the probability that Template:Mvar falls in the range Template:Closed-closed.

Two closely related functions are the complementary error function $e r f c : ℂ \to ℂ$ is defined as

$erfc (z) = 1 - \erf (z),$

and the imaginary error function $e r f i : ℂ \to ℂ$ is defined as

$erfi (z) = - i \erf (i z),$

where Template:Mvar is the imaginary unit.

Name

The name "error function" and its abbreviation Template:Math were proposed by J. W. L. Glaisher in 1871 on account of its connection with "the theory of Probability, and notably the theory of Errors."^[3] The error function complement was also discussed by Glaisher in a separate publication in the same year.^[4] For the "law of facility" of errors whose density is given by $f (x) = {(\frac{c}{π})}^{1 / 2} e^{- c x^{2}}$ (the normal distribution), Glaisher calculates the probability of an error lying between Template:Mvar and Template:Mvar as: ${(\frac{c}{π})}^{\frac{1}{2}} \int_{p}^{q} e^{- c x^{2}} d x = \frac{1}{2} (\erf (q \sqrt{c}) - \erf (p \sqrt{c})) .$

Plot of the error function erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

Applications

When the results of a series of measurements are described by a normal distribution with standard deviation Template:Mvar and expected value 0, then Template:Math is the probability that the error of a single measurement lies between Template:Math and Template:Math, for positive Template:Mvar. This is useful, for example, in determining the bit error rate of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

The error function and its approximations can be used to estimate results that hold with high probability or with low probability. Given a random variable Template:Math (a normal distribution with mean Template:Mvar and standard deviation Template:Mvar) and a constant Template:Math, it can be shown via integration by substitution: $\begin{aligned} \Pr [X \leq L] & = \frac{1}{2} + \frac{1}{2} \erf (\frac{L - μ}{\sqrt{2} σ}) \\ \approx A \exp (- B {(\frac{L - μ}{σ})}^{2}) \end{aligned}$

where Template:Mvar and Template:Mvar are certain numeric constants. If Template:Mvar is sufficiently far from the mean, specifically Template:Math, then:

$\Pr [X \leq L] \leq A \exp (- B \ln (k)) = \frac{A}{k^{B}}$

so the probability goes to 0 as Template:Math.

The probability for Template:Mvar being in the interval Template:Closed-closed can be derived as $\begin{aligned} \Pr [L_{a} \leq X \leq L_{b}] & = \int_{L_{a}}^{L_{b}} \frac{1}{\sqrt{2 π} σ} \exp (- \frac{(x - μ)^{2}}{2 σ^{2}}) d x \\ = \frac{1}{2} (\erf (\frac{L_{b} - μ}{\sqrt{2} σ}) - \erf (\frac{L_{a} - μ}{\sqrt{2} σ})) . \end{aligned}$

Properties

Template:Multiple image

The property Template:Math means that the error function is an odd function. This directly results from the fact that the integrand Template:Math 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 Template:Mvar: $\erf (\overline{z}) = \overline{\erf (z)}$ where $\overline{z}$ denotes the complex conjugate of $z$ .

The integrand Template:Math and Template:Math are shown in the complex Template:Mvar-plane in the figures at right with domain coloring.

The error function at Template:Math is exactly 1 (see Gaussian integral). At the real axis, Template:Math approaches unity at Template:Math and −1 at Template:Math. At the imaginary axis, it tends to Template:Math.

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges. For Template:Math, however, cancellation of leading terms makes the Taylor expansion unpractical.

The defining integral cannot be evaluated in closed form in terms of elementary functions (see Liouville's theorem), but by expanding the integrand Template:Math into its Maclaurin series and integrating term by term, one obtains the error function's Maclaurin series as: $\begin{aligned} \erf (z) & = \frac{2}{\sqrt{π}} \sum_{n = 0}^{\infty} \frac{(- 1)^{n} z^{2 n + 1}}{n! (2 n + 1)} \\ = \frac{2}{\sqrt{π}} (z - \frac{z^{3}}{3} + \frac{z^{5}}{10} - \frac{z^{7}}{42} + \frac{z^{9}}{216} - \dots) \end{aligned}$ which holds for every complex number Template:Mvar. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful: $\begin{aligned} \erf (z) & = \frac{2}{\sqrt{π}} \sum_{n = 0}^{\infty} (z \prod_{k = 1}^{n} \frac{- (2 k - 1) z^{2}}{k (2 k + 1)}) \\ = \frac{2}{\sqrt{π}} \sum_{n = 0}^{\infty} \frac{z}{2 n + 1} \prod_{k = 1}^{n} \frac{- z^{2}}{k} \end{aligned}$ because Template:Math expresses the multiplier to turn the Template:Mvarth term into the Template:Mathth term (considering Template:Mvar as the first term).

The imaginary error function has a very similar Maclaurin series, which is: $\begin{aligned} erfi (z) & = \frac{2}{\sqrt{π}} \sum_{n = 0}^{\infty} \frac{z^{2 n + 1}}{n! (2 n + 1)} \\ = \frac{2}{\sqrt{π}} (z + \frac{z^{3}}{3} + \frac{z^{5}}{10} + \frac{z^{7}}{42} + \frac{z^{9}}{216} + \dots) \end{aligned}$ which holds for every complex number Template:Mvar.

Derivative and integral

The derivative of the error function follows immediately from its definition: $\frac{d}{d z} \erf (z) = \frac{2}{\sqrt{π}} e^{- z^{2}} .$ From this, the derivative of the imaginary error function is also immediate: $\frac{d}{d z} erfi (z) = \frac{2}{\sqrt{π}} e^{z^{2}} .$ An antiderivative of the error function, obtainable by integration by parts, is $z \erf (z) + \frac{e^{- z^{2}}}{\sqrt{π}} + C .$ An antiderivative of the imaginary error function, also obtainable by integration by parts, is $z erfi (z) - \frac{e^{z^{2}}}{\sqrt{π}} + C .$ Higher order derivatives are given by $\erf^{(k)} (z) = \frac{2 (- 1)^{k - 1}}{\sqrt{π}} H_{k - 1} (z) e^{- z^{2}} = \frac{2}{\sqrt{π}} \frac{d^{k - 1}}{d z^{k - 1}} (e^{- z^{2}}), k = 1, 2, \dots$ where Template:Mvar are the physicists' Hermite polynomials.^[5]

Bürmann series

An expansion,^[6] which converges more rapidly for all real values of Template:Mvar than a Taylor expansion, is obtained by using Hans Heinrich Bürmann's theorem:^[7] $\begin{aligned} \erf (x) & = \frac{2}{\sqrt{π}} sgn (x) \cdot \sqrt{1 - e^{- x^{2}}} (1 - \frac{1}{12} (1 - e^{- x^{2}}) - \frac{7}{480} {(1 - e^{- x^{2}})}^{2} - \frac{5}{896} {(1 - e^{- x^{2}})}^{3} - \frac{787}{276480} {(1 - e^{- x^{2}})}^{4} - \dots) \\ = \frac{2}{\sqrt{π}} sgn (x) \cdot \sqrt{1 - e^{- x^{2}}} (\frac{\sqrt{π}}{2} + \sum_{k = 1}^{\infty} c_{k} e^{- k x^{2}}) . \end{aligned}$ where Template:Math is the sign function. By keeping only the first two coefficients and choosing Template:Math and Template:Math, the resulting approximation shows its largest relative error at Template:Math, where it is less than 0.0034361: $\erf (x) \approx \frac{2}{\sqrt{π}} sgn (x) \cdot \sqrt{1 - e^{- x^{2}}} (\frac{\sqrt{π}}{2} + \frac{31}{200} e^{- x^{2}} - \frac{341}{8000} e^{- 2 x^{2}}) .$

Inverse functions

File:Mplwp erf inv.svg

Inverse error function

Given a complex number Template:Mvar, there is not a unique complex number Template:Mvar satisfying Template:Math, so a true inverse function would be multivalued. However, for Template:Math, there is a unique real number denoted Template:Math satisfying $\erf (\erf^{- 1} (x)) = x .$

The inverse error function is usually defined with domain Template:Open-open, and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk Template:Math of the complex plane, using the Maclaurin series^[8] $\erf^{- 1} (z) = \sum_{k = 0}^{\infty} \frac{c_{k}}{2 k + 1} {(\frac{\sqrt{π}}{2} z)}^{2 k + 1},$ where Template:Math and $\begin{aligned} c_{k} & = \sum_{m = 0}^{k - 1} \frac{c_{m} c_{k - 1 - m}}{(m + 1) (2 m + 1)} \\ = {1, 1, \frac{7}{6}, \frac{127}{90}, \frac{4369}{2520}, \frac{34807}{16200}, \dots} . \end{aligned}$

So we have the series expansion (common factors have been canceled from numerators and denominators): $\erf^{- 1} (z) = \frac{\sqrt{π}}{2} (z + \frac{π}{12} z^{3} + \frac{7 π^{2}}{480} z^{5} + \frac{127 π^{3}}{40320} z^{7} + \frac{4369 π^{4}}{5806080} z^{9} + \frac{34807 π^{5}}{182476800} z^{11} + \dots) .$ (After cancellation the numerator and denominator values in Template:Oeis and Template:Oeis respectively; without cancellation the numerator terms are values in Template:Oeis.) The error function's value at Template:Math is equal to Template:Math.

For Template:Math, we have Template:Math.

The inverse complementary error function is defined as ${erfc}^{- 1} (1 - z) = \erf^{- 1} (z) .$ For real Template:Mvar, there is a unique real number Template:Math satisfying Template:Math. The inverse imaginary error function is defined as Template:Math.^[9]

For any real x, Newton's method can be used to compute Template:Math, and for Template:Math, the following Maclaurin series converges: ${erfi}^{- 1} (z) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} c_{k}}{2 k + 1} {(\frac{\sqrt{π}}{2} z)}^{2 k + 1},$ where Template:Math 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 Template:Mvar is $\begin{aligned} erfc (x) & = \frac{e^{- x^{2}}}{x \sqrt{π}} (1 + \sum_{n = 1}^{\infty} (- 1)^{n} \frac{1 \cdot 3 \cdot 5 \dots (2 n - 1)}{{(2 x^{2})}^{n}}) \\ = \frac{e^{- x^{2}}}{x \sqrt{π}} \sum_{n = 0}^{\infty} (- 1)^{n} \frac{(2 n - 1)!!}{{(2 x^{2})}^{n}}, \end{aligned}$ where Template:Math is the double factorial of Template:Math, which is the product of all odd numbers up to Template:Math. This series diverges for every finite Template:Mvar, and its meaning as asymptotic expansion is that for any integer Template:Math one has $erfc (x) = \frac{e^{- x^{2}}}{x \sqrt{π}} \sum_{n = 0}^{N - 1} (- 1)^{n} \frac{(2 n - 1)!!}{{(2 x^{2})}^{n}} + R_{N} (x)$ where the remainder is $R_{N} (x) : = \frac{(- 1)^{N} (2 N - 1)!!}{\sqrt{π} \cdot 2^{N - 1}} \int_{x}^{\infty} t^{- 2 N} e^{- t^{2}} d t,$ which follows easily by induction, writing $e^{- t^{2}} = - \frac{1}{2 t} \frac{d}{d t} e^{- t^{2}}$ and integrating by parts.

The asymptotic behavior of the remainder term, in Landau notation, is $R_{N} (x) = O (x^{- (1 + 2 N)} e^{- x^{2}})$ as Template:Math. This can be found by $R_{N} (x) \propto \int_{x}^{\infty} t^{- 2 N} e^{- t^{2}} d t = e^{- x^{2}} \int_{0}^{\infty} (t + x)^{- 2 N} e^{- t^{2} - 2 t x} d t \leq e^{- x^{2}} \int_{0}^{\infty} x^{- 2 N} e^{- 2 t x} d t \propto x^{- (1 + 2 N)} e^{- x^{2}} .$ For large enough values of Template:Mvar, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of Template:Math (while for not too large values of Template:Mvar, 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 Laplace:^[10]^[11] $erfc (z) = \frac{z}{\sqrt{π}} e^{- z^{2}} \frac{1}{z^{2} + \frac{a_{1}}{1 + \frac{a_{2}}{z^{2} + \frac{a_{3}}{1 + \dots}}}}, a_{m} = \frac{m}{2} .$

Factorial series

The inverse factorial series: $\begin{aligned} erfc (z) & = \frac{e^{- z^{2}}}{\sqrt{π} z} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} Q_{n}}{{(z^{2} + 1)}^{\bar{n}}} \\ = \frac{e^{- z^{2}}}{\sqrt{π} z} [1 - \frac{1}{2} \frac{1}{(z^{2} + 1)} + \frac{1}{4} \frac{1}{(z^{2} + 1) (z^{2} + 2)} - \dots] \end{aligned}$ converges for Template:Math. Here $\begin{aligned} Q_{n} & \overset{def}{=} \frac{1}{Γ (\frac{1}{2})} \int_{0}^{\infty} τ (τ - 1) \dots (τ - n + 1) τ^{- \frac{1}{2}} e^{- τ} d τ \\ = \sum_{k = 0}^{n} {(\frac{1}{2})}^{\bar{k}} s (n, k), \end{aligned}$ Template:Math denotes the rising factorial, and Template:Math denotes a signed Stirling number of the first kind.^[12]^[13] There also exists a representation by an infinite sum containing the double factorial: $\erf (z) = \frac{2}{\sqrt{π}} \sum_{n = 0}^{\infty} \frac{(- 2)^{n} (2 n - 1)!!}{(2 n + 1)!} z^{2 n + 1}$

Bounds and Numerical approximations

Approximation with elementary functions

Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are: $\erf (x) \approx 1 - \frac{1}{{(1 + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4})}^{4}}, x \geq 0$ (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math $\erf (x) \approx 1 - (a_{1} t + a_{2} t^{2} + a_{3} t^{3}) e^{- x^{2}}, t = \frac{1}{1 + p x}, x \geq 0$ (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math $\erf (x) \approx 1 - \frac{1}{{(1 + a_{1} x + a_{2} x^{2} + \dots + a_{6} x^{6})}^{16}}, x \geq 0$ (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math $\erf (x) \approx 1 - (a_{1} t + a_{2} t^{2} + \dots + a_{5} t^{5}) e^{- x^{2}}, t = \frac{1}{1 + p x}$ (maximum error: Template:Val) Template:Pb where Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math Template:Pb All of these approximations are valid for Template:Math. To use these approximations for negative Template:Mvar, use the fact that Template:Math is an odd function, so Template:Math.
Exponential bounds and a pure exponential approximation for the complementary error function are given by^[14] $\begin{aligned} erfc (x) & \leq \frac{1}{2} e^{- 2 x^{2}} + \frac{1}{2} e^{- x^{2}} \leq e^{- x^{2}}, & x & > 0 \\ erfc (x) & \approx \frac{1}{6} e^{- x^{2}} + \frac{1}{2} e^{- \frac{4}{3} x^{2}}, & x & > 0 . \end{aligned}$
The above have been generalized to sums of Template:Mvar exponentials^[15] with increasing accuracy in terms of Template:Mvar so that Template:Math can be accurately approximated or bounded by Template:Math, where $\tilde{Q} (x) = \sum_{n = 1}^{N} a_{n} e^{- b_{n} x^{2}} .$ In particular, there is a systematic methodology to solve the numerical coefficients Template:Math that yield a minimax approximation or bound for the closely related Q-function: Template:Math, Template:Math, or Template:Math for Template:Math. The coefficients Template:Math for many variations of the exponential approximations and bounds up to Template:Math have been released to open access as a comprehensive dataset.^[16]
A tight approximation of the complementary error function for Template:Math is given by Karagiannidis & Lioumpas (2007)^[17] who showed for the appropriate choice of parameters Template:Math that $erfc (x) \approx \frac{(1 - e^{- A x}) e^{- x^{2}}}{B \sqrt{π} x} .$ They determined Template:Math, which gave a good approximation for all Template:Math. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.^[18]
A single-term lower bound is^[19] $erfc (x) \geq \sqrt{\frac{2 e}{π}} \frac{\sqrt{β - 1}}{β} e^{- β x^{2}}, x \geq 0, β > 1,$ where the parameter Template:Mvar can be picked to minimize error on the desired interval of approximation.
Another approximation is given by Sergei Winitzki using his "global Padé approximations":^[20]^[21]Template:Rp $\erf (x) \approx sgn x \cdot \sqrt{1 - \exp (- x^{2} \frac{\frac{4}{π} + a x^{2}}{1 + a x^{2}})}$ where $a = \frac{8 (π - 3)}{3 π (4 - π)} \approx 0.140012 .$ This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the relative error is less than 0.00035 for all real Template:Mvar. Using the alternate value Template:Math reduces the maximum relative error to about 0.00013.^[22] Template:Pb This approximation can be inverted to obtain an approximation for the inverse error function: $\erf^{- 1} (x) \approx sgn x \cdot \sqrt{\sqrt{{(\frac{2}{π a} + \frac{\ln (1 - x^{2})}{2})}^{2} - \frac{\ln (1 - x^{2})}{a}} - (\frac{2}{π a} + \frac{\ln (1 - x^{2})}{2})} .$
An approximation with a maximal error of Template:Val for any real argument is:^[23] $\erf (x) = {\begin{cases} 1 - τ & x \geq 0 \\ τ - 1 & x < 0 \end{cases}$ with $\begin{aligned} τ & = t \cdot \exp (- x^{2} - 1.26551223 + 1.00002368 t + 0.37409196 t^{2} + 0.09678418 t^{3} - 0.18628806 t^{4} \\ + 0.27886807 t^{5} - 1.13520398 t^{6} + 1.48851587 t^{7} - 0.82215223 t^{8} + 0.17087277 t^{9}) \end{aligned}$ and $t = \frac{1}{1 + \frac{1}{2} | x |} .$
An approximation of $erfc$ with a maximum relative error less than $2^{- 53}$ $(\approx 1.1 \times 1 0^{- 16})$ in absolute value is:^[24] for $x \geq 0$ , $\begin{aligned} erfc (x) & = (\frac{0.56418958354775629}{x + 2.06955023132914151}) (\frac{x^{2} + 2.71078540045147805 x + 5.80755613130301624}{x^{2} + 3.47954057099518960 x + 12.06166887286239555}) \\ (\frac{x^{2} + 3.47469513777439592 x + 12.07402036406381411}{x^{2} + 3.72068443960225092 x + 8.44319781003968454}) (\frac{x^{2} + 4.00561509202259545 x + 9.30596659485887898}{x^{2} + 3.90225704029924078 x + 6.36161630953880464}) \\ (\frac{x^{2} + 5.16722705817812584 x + 9.12661617673673262}{x^{2} + 4.03296893109262491 x + 5.13578530585681539}) (\frac{x^{2} + 5.95908795446633271 x + 9.19435612886969243}{x^{2} + 4.11240942957450885 x + 4.48640329523408675}) e^{- x^{2}} \end{aligned}$ and for $x < 0$ $erfc (x) = 2 - erfc (- x)$
A simple approximation for real-valued arguments could be done through Hyperbolic functions: $\erf (x) \approx z (x) = \tanh (\frac{2}{\sqrt{π}} (x + \frac{11}{123} x^{3}))$ which keeps the absolute difference $| \erf (x) - z (x) | < 0.000358, \forall x$ .
Since the error function and the Gaussian Q-function are closely related through the identity $erfc (x) = 2 Q (\sqrt{2} x)$ or equivalently $Q (x) = \frac{1}{2} erfc (\frac{x}{\sqrt{2}})$ , bounds developed for the Q-function can be adapted to approximate the complementary error function. A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments $x \in [0, \infty)$ was introduced by Abreu (2012)^[25] based on a simple algebraic expression with only two exponential terms: $Q (x) \geq \frac{1}{12} e^{- x^{2}} + \frac{1}{\sqrt{2 π} (x + 1)} e^{- x^{2} / 2}, x \geq 0,$ and $Q (x) \leq \frac{1}{50} e^{- x^{2}} + \frac{1}{2 (x + 1)} e^{- x^{2} / 2}, x \geq 0 .$ These bounds stem from a unified form $Q_{B} (x; a, b) = \frac{\exp (- x^{2})}{a} + \frac{\exp (- x^{2} / 2)}{b (x + 1)},$ where the parameters $a$ and $b$ are selected to ensure the bounding properties: for the lower bound, $a_{L} = 12$ and $b_{L} = \sqrt{2 π}$ , and for the upper bound, $a_{U} = 50$ and $b_{U} = 2$ . These expressions maintain simplicity and tightness, providing a practical trade-off between accuracy and ease of computation. They are particularly valuable in theoretical contexts, such as communication theory over fading channels, where both functions frequently appear. Additionally, the original Q-function bounds can be extended to $Q^{n} (x)$ for positive integers $n$ via the binomial theorem, suggesting potential adaptability for powers of $erfc (x)$ , though this is less commonly required in error function applications.

Table of values

Script error: No such module "labelled list hatnote".

Template:Math	Template:Math	Template:Math
0	Template:Val	Template:Val
0.02	Template:Val	Template:Val
0.04	Template:Val	Template:Val
0.06	Template:Val	Template:Val
0.08	Template:Val	Template:Val
0.1	Template:Val	Template:Val
0.2	Template:Val	Template:Val
0.3	Template:Val	Template:Val
0.4	Template:Val	Template:Val
0.5	Template:Val	Template:Val
0.6	Template:Val	Template:Val
0.7	Template:Val	Template:Val
0.8	Template:Val	Template:Val
0.9	Template:Val	Template:Val
1	Template:Val	Template:Val
1.1	Template:Val	Template:Val
1.2	Template:Val	Template:Val
1.3	Template:Val	Template:Val
1.4	Template:Val	Template:Val
1.5	Template:Val	Template:Val
1.6	Template:Val	Template:Val
1.7	Template:Val	Template:Val
1.8	Template:Val	Template:Val
1.9	Template:Val	Template:Val
2	Template:Val	Template:Val
2.1	Template:Val	Template:Val
2.2	Template:Val	Template:Val
2.3	Template:Val	Template:Val
2.4	Template:Val	Template:Val
2.5	Template:Val	Template:Val
3	Template:Val	Template:Val
3.5	Template:Val	Template:Val

Related functions

Complementary error function

The complementary error function, denoted Template:Math, is defined as

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

$\begin{aligned} erfc (x) & = 1 - \erf (x) \\ = \frac{2}{\sqrt{π}} \int_{x}^{\infty} e^{- t^{2}} d t \\ = e^{- x^{2}} erfcx (x), \end{aligned}$ which also defines Template:Math, the scaled complementary error function^[26] (which can be used instead of Template:Math to avoid arithmetic underflow^[26]^[27]). Another form of Template:Math for Template:Math is known as Craig's formula, after its discoverer:^[28] $erfc (x ∣ x \geq 0) = \frac{2}{π} \int_{0}^{\frac{π}{2}} \exp (- \frac{x^{2}}{\sin^{2} θ}) d θ .$ This expression is valid only for positive values of Template:Mvar, but it can be used in conjunction with Template:Math to obtain Template:Math for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the Template:Math of the sum of two non-negative variables is as follows:^[29] $erfc (x + y ∣ x, y \geq 0) = \frac{2}{π} \int_{0}^{\frac{π}{2}} \exp (- \frac{x^{2}}{\sin^{2} θ} - \frac{y^{2}}{\cos^{2} θ}) d θ .$

Imaginary error function

The imaginary error function, denoted Template:Math, is defined as

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

$\begin{aligned} erfi (x) & = - i \erf (i x) \\ = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{t^{2}} d t \\ = \frac{2}{\sqrt{π}} e^{x^{2}} D (x), \end{aligned}$ where Template:Math is the Dawson function (which can be used instead of Template:Math to avoid arithmetic overflow^[26]).

Despite the name "imaginary error function", Template:Math is real when Template:Mvar is real.

When the error function is evaluated for arbitrary complex arguments Template:Mvar, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: $w (z) = e^{- z^{2}} erfc (- i z) = erfcx (- i z) .$

Cumulative distribution function

The error function is essentially identical to the standard normal cumulative distribution function, denoted Template:Math, also named Template:Math by some software languagesScript error: No such module "Unsubst"., as they differ only by scaling and translation. Indeed,

the normal cumulative distribution function plotted in the complex plane

$\begin{aligned} Φ (x) & = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} e^{\frac{- t^{2}}{2}} d t \\ = \frac{1}{2} (1 + \erf (\frac{x}{\sqrt{2}})) \\ = \frac{1}{2} erfc (- \frac{x}{\sqrt{2}}) \end{aligned}$ or rearranged for Template:Math and Template:Math: $\begin{aligned} \erf (x) & = 2 Φ (x \sqrt{2}) - 1 \\ erfc (x) & = 2 Φ (- x \sqrt{2}) \\ = 2 (1 - Φ (x \sqrt{2})) . \end{aligned}$

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 $\begin{aligned} Q (x) & = \frac{1}{2} - \frac{1}{2} \erf (\frac{x}{\sqrt{2}}) \\ = \frac{1}{2} erfc (\frac{x}{\sqrt{2}}) . \end{aligned}$

The inverse of Template:Math is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as $probit (p) = Φ^{- 1} (p) = \sqrt{2} \erf^{- 1} (2 p - 1) = - \sqrt{2} {erfc}^{- 1} (2 p) .$

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): $\erf (x) = \frac{2 x}{\sqrt{π}} M (\frac{1}{2}, \frac{3}{2}, - x^{2}) .$

It has a simple expression in terms of the Fresnel integral.Template:Elucidate

In terms of the regularized gamma function Template:Mvar and the incomplete gamma function, $\erf (x) = sgn (x) \cdot P (\frac{1}{2}, x^{2}) = \frac{sgn (x)}{\sqrt{π}} γ (\frac{1}{2}, x^{2}) .$ Template:Math is the sign function.

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by^[30] $\begin{aligned} i^{n} erfc (z) & = \int_{z}^{\infty} i^{n - 1} erfc (ζ) d ζ \\ i^{0} erfc (z) & = erfc (z) \\ i^{1} erfc (z) & = ierfc (z) = \frac{1}{\sqrt{π}} e^{- z^{2}} - z erfc (z) \\ i^{2} erfc (z) & = \frac{1}{4} (erfc (z) - 2 z ierfc (z)) \end{aligned}$

The general recurrence formula is $2 n \cdot i^{n} erfc (z) = i^{n - 2} erfc (z) - 2 z \cdot i^{n - 1} erfc (z)$

They have the power series $i^{n} erfc (z) = \sum_{j = 0}^{\infty} \frac{(- z)^{j}}{2^{n - j} j! Γ (1 + \frac{n - j}{2})},$ from which follow the symmetry properties $i^{2 m} erfc (- z) = - i^{2 m} erfc (z) + \sum_{q = 0}^{m} \frac{z^{2 q}}{2^{2 (m - q) - 1} (2 q)! (m - q)!}$ and $i^{2 m + 1} erfc (- z) = i^{2 m + 1} erfc (z) + \sum_{q = 0}^{m} \frac{z^{2 q + 1}}{2^{2 (m - q) - 1} (2 q + 1)! (m - q)!} .$

Implementations

As real function of a real argument

In POSIX-compliant operating systems, the header math.h shall declare and the mathematical library libm shall provide the functions erf and erfc (double precision) as well as their single precision and extended precision counterparts erff, erfl and erfcf, erfcl.^[31]
The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.^[32]

As complex function of a complex argument

libcerf, numeric C library for complex error functions, provides the complex functions cerf, cerfc, cerfcx and the real functions erfi, erfcx with approximately 13–14 digits precision, based on the Faddeeva function as implemented in the MIT Faddeeva Package

References

Template:Reflist

External links

A Table of Integrals of the Error Functions

Template:Nonelementary Integral Template:Authority control

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Error function: Difference between revisions

Latest revision as of 13:25, 22 June 2025

Contents

Name

Applications

Properties

Taylor series

Derivative and integral

Bürmann series

Inverse functions

Asymptotic expansion

Continued fraction expansion

Factorial series

Bounds and Numerical approximations

Approximation with elementary functions

Table of values

Related functions

Complementary error function

Imaginary error function

Cumulative distribution function

Iterated integrals of the complementary error function

Implementations

As real function of a real argument

As complex function of a complex argument

References

Further reading

External links

Navigation menu

@@ Line 3: / Line 3: @@
 {{Distinguish|Loss function}}
 In mathematics, the '''error function''' (also called the '''Gauss error function'''), often denoted by '''{{math|erf}}''', is a function <math>\mathrm{erf}: \mathbb{C} \to \mathbb{C}</math> defined as:<ref>{{cite book|last =Andrews|first = Larry C.|url = https://books.google.com/books?id=2CAqsF-RebgC&pg=PA110 |title = Special functions of mathematics for engineers|page = 110|publisher = SPIE Press |date= 1998|isbn = 9780819426161}}</ref>
-<math display="block">\operatorname{erf} z = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,\mathrm dt.</math>
+<math display="block">\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,\mathrm dt.</math>
 {{Infobox mathematical function
 | name = Error function
@@ Line 10: / Line 10: @@
 | imagealt = Plot of the error function over real numbers
 | caption = Plot of the error function over real numbers
-| general_definition = <math>\operatorname{erf} z = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,\mathrm dt</math>
+| general_definition = <math>\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,\mathrm dt</math>
 | fields_of_application = Probability, thermodynamics, digital communications
 | domain = <math>\mathbb{C}</math>
@@ Line 16: / Line 16: @@
 | parity = Odd
 | root = 0
-| derivative = <math>\frac{\mathrm d}{\mathrm dz}\operatorname{erf} z = \frac{2}{\sqrt\pi} e^{-z^2} </math>
+| derivative = <math>\frac{\mathrm d}{\mathrm dz}\operatorname{erf}(z) = \frac{2}{\sqrt\pi} e^{-z^2} </math>
-| antiderivative = <math>\int \operatorname{erf} z\,dz = z \operatorname{erf} z + \frac{e^{-z^2}}{\sqrt\pi} + C</math>
+| antiderivative = <math>\int \operatorname{erf}(z)\,dz = z \operatorname{erf}(z) + \frac{e^{-z^2}}{\sqrt\pi} + C</math>
-| taylor_series = <math>\operatorname{erf} z = \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \frac{z^{2n+1}}{n!}</math>
+| taylor_series = <math>\operatorname{erf}(z) = \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \frac{z^{2n+1}}{n!}</math>
 }}
@@ Line 27: / Line 27: @@
 This [[nonelementary integral]] is a [[sigmoid function|sigmoid]] function that occurs often in [[probability]], [[statistics]], and [[partial differential equation]]s.
-In statistics, for non-negative real values of {{mvar|x}}, the error function has the following interpretation: for a real [[random variable]] {{mvar|Y}} that is [[normal distribution|normally distributed]] with [[mean]] 0 and [[standard deviation]] <math>\frac{1}{\sqrt{2}}</math>, {{math|erf ''x''}} is the probability that {{mvar|Y}} falls in the range {{closed-closed|−''x'', ''x''}}.
+In statistics, for non-negative real values of {{mvar|x}}, the error function has the following interpretation: for a real [[random variable]] {{mvar|Y}} that is [[normal distribution|normally distributed]] with [[mean]] 0 and [[standard deviation]] <math>\frac{1}{\sqrt{2}}</math>, {{math|erf(''x'')}} is the probability that {{mvar|Y}} falls in the range {{closed-closed|−''x'', ''x''}}.
 Two closely related functions are the '''complementary error function''' <math>\mathrm{erfc}: \mathbb{C} \to \mathbb{C}</math> is defined as
-<math display="block">\operatorname{erfc} z = 1 - \operatorname{erf} z,</math>
+<math display="block">\operatorname{erfc}(z) = 1 - \operatorname{erf}(z),</math>
 and the '''imaginary error function''' <math>\mathrm{erfi}: \mathbb{C} \to \mathbb{C}</math> is defined as
-<math display="block">\operatorname{erfi} z = -i\operatorname{erf} iz,</math>
+<math display="block">\operatorname{erfi}(z) = -i\operatorname{erf}(iz),</math>
 where {{mvar|i}} is the [[imaginary unit]].
@@ Line 45: / Line 45: @@
 (the [[normal distribution]]), Glaisher calculates the probability of an error lying between {{mvar|p}} and {{mvar|q}} as:
 <math display="block">\left(\frac{c}{\pi}\right)^\frac{1}{2} \int_p^qe^{-cx^2}\,\mathrm dx = \tfrac{1}{2}\left(\operatorname{erf} \left(q\sqrt{c}\right) -\operatorname{erf} \left(p\sqrt{c}\right)\right).</math>
-[[File:Plot of the error function Erf(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 error function Erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the error function Erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
+[[File:Plot of the error function Erf(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 error function erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the error function erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
 ==Applications==
@@ Line 54: / Line 54: @@
 The error function and its approximations can be used to estimate results that hold [[with high probability]] or with low probability. Given a random variable {{math|''X'' ~ Norm[''μ'',''σ'']}} (a normal distribution with mean {{mvar|μ}} and standard deviation {{mvar|σ}}) and a constant {{math|''L'' > ''μ''}}, it can be shown via integration by substitution:
 <math display="block">\begin{align}
-\Pr[X\leq L] &= \frac{1}{2} + \frac{1}{2} \operatorname{erf}\frac{L-\mu}{\sqrt{2}\sigma} \\
+\Pr[X\leq L] &= \frac{1}{2} + \frac{1}{2} \operatorname{erf}\left(\frac{L-\mu}{\sqrt{2}\sigma}\right) \\
 &\approx A \exp \left(-B \left(\frac{L-\mu}{\sigma}\right)^2\right)
 \end{align}</math>
-where {{mvar|A}} and {{mvar|B}} are certain numeric constants. If {{mvar|L}} is sufficiently far from the mean, specifically {{math|''μ'' − ''L'' ≥ ''σ''{{sqrt|ln ''k''}}}}, then:
+where {{mvar|A}} and {{mvar|B}} are certain numeric constants. If {{mvar|L}} is sufficiently far from the mean, specifically {{math|''μ'' − ''L'' ≥ ''σ''{{sqrt|ln(''k'')}}}}, then:
-<math display="block">\Pr[X\leq L] \leq A \exp (-B \ln{k}) = \frac{A}{k^B}</math>
+<math display="block">\Pr[X\leq L] \leq A \exp (-B \ln(k)) = \frac{A}{k^B}</math>
 so the probability goes to 0 as {{math|''k'' → ∞}}.
@@ Line 67: / Line 67: @@
 <math display="block">\begin{align}
 \Pr[L_a\leq X \leq L_b] &= \int_{L_a}^{L_b} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \,\mathrm dx \\
-&= \frac{1}{2}\left(\operatorname{erf}\frac{L_b-\mu}{\sqrt{2}\sigma} - \operatorname{erf}\frac{L_a-\mu}{\sqrt{2}\sigma}\right).\end{align}</math>
+&= \frac{1}{2}\left(\operatorname{erf}\left(\frac{L_b-\mu}{\sqrt{2}\sigma}\right) - \operatorname{erf}\left(\frac{L_a-\mu}{\sqrt{2}\sigma}\right)\right).\end{align}</math>
 ==Properties==
@@ Line 77: / Line 77: @@
   | caption1  = Integrand {{math|exp(−''z''<sup>2</sup>)}}
   | image2    = ComplexErfz.png
-  | caption2  = {{math|erf ''z''}}
+  | 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).
+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>
+<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 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''∞}}.
@@ Line 97: / Line 97: @@
 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
+\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)
@@ Line 105: / Line 105: @@
 For iterative calculation of the above series, the following alternative formulation may be useful:
 <math display="block">\begin{align}
-\operatorname{erf} z
+\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}
@@ Line 113: / Line 113: @@
 The imaginary error function has a very similar Maclaurin series, which is:
 <math display="block">\begin{align}
-\operatorname{erfi} z
+\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)
@@ Line 121: / Line 121: @@
 ===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>
+<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>
+<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>
+<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>
+<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>
+<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>
@@ Line 136: / Line 136: @@
 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
+\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( 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).
+&= \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>
+<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
+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>
+<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>
+<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}
@@ Line 158: / Line 158: @@
 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>
+<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''}}.
+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>
+<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 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:
+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>
+<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.
@@ Line 174: / Line 174: @@
 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]
+\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>
+<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>
@@ Line 193: / Line 193: @@
 ===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>
+<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
+\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]
@@ Line 211: / Line 211: @@
 {{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>
+<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>
 == Bounds and Numerical approximations ==
@@ Line 219: / Line 219: @@
 <li>
 [[Abramowitz and Stegun]] give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are:
-<math display="block">\operatorname{erf} x \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\right)^4}, \qquad x \geq 0</math>
+<math display="block">\operatorname{erf}(x) \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\right)^4}, \qquad x \geq 0</math>
 (maximum error: {{val|5e-4}})
 {{pb}}
 where {{math|''a''<sub>1</sub> {{=}} 0.278393}}, {{math|''a''<sub>2</sub> {{=}} 0.230389}}, {{math|''a''<sub>3</sub> {{=}} 0.000972}}, {{math|''a''<sub>4</sub> {{=}} 0.078108}}
-<math display="block">\operatorname{erf} x \approx 1 - \left(a_1t + a_2t^2 + a_3t^3\right)e^{-x^2},\quad t=\frac{1}{1 + px}, \qquad x \geq 0</math>
+<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + a_3t^3\right)e^{-x^2},\quad t=\frac{1}{1 + px}, \qquad x \geq 0</math>
 (maximum error: {{val|2.5e-5}})
 {{pb}}
 where {{math|''p'' {{=}} 0.47047}}, {{math|''a''<sub>1</sub> {{=}} 0.3480242}}, {{math|''a''<sub>2</sub> {{=}} −0.0958798}}, {{math|''a''<sub>3</sub> {{=}} 0.7478556}}
-<math display="block">\operatorname{erf} x \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + \cdots + a_6x^6\right)^{16}}, \qquad x \geq 0</math>
+<math display="block">\operatorname{erf}(x) \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + \cdots + a_6x^6\right)^{16}}, \qquad x \geq 0</math>
 (maximum error: {{val|3e-7}})
 {{pb}}
 where {{math|''a''<sub>1</sub> {{=}} 0.0705230784}}, {{math|''a''<sub>2</sub> {{=}} 0.0422820123}}, {{math|''a''<sub>3</sub> {{=}} 0.0092705272}}, {{math|''a''<sub>4</sub> {{=}} 0.0001520143}}, {{math|''a''<sub>5</sub> {{=}} 0.0002765672}}, {{math|''a''<sub>6</sub> {{=}} 0.0000430638}}
-<math display="block">\operatorname{erf} x \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5\right)e^{-x^2},\quad t = \frac{1}{1 + px}</math>
+<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5\right)e^{-x^2},\quad t = \frac{1}{1 + px}</math>
 (maximum error: {{val|1.5e-7}})
 {{pb}}
 where {{math|''p'' {{=}} 0.3275911}}, {{math|''a''<sub>1</sub> {{=}} 0.254829592}}, {{math|''a''<sub>2</sub> {{=}} −0.284496736}}, {{math|''a''<sub>3</sub> {{=}} 1.421413741}}, {{math|''a''<sub>4</sub> {{=}} −1.453152027}}, {{math|''a''<sub>5</sub> {{=}} 1.061405429}}
 {{pb}}
-All of these approximations are valid for {{math|''x'' ≥ 0}}.  To use these approximations for negative {{mvar|x}}, use the fact that {{math|erf ''x''}} is an odd function, so {{math|erf ''x'' {{=}} −erf(−''x'')}}.
+All of these approximations are valid for {{math|''x'' ≥ 0}}.  To use these approximations for negative {{mvar|x}}, use the fact that {{math|erf(''x'')}} is an odd function, so {{math|erf(''x'') {{=}} −erf(−''x'')}}.
 </li>
@@ Line 245: / Line 245: @@
 Exponential bounds and a pure exponential approximation for the complementary error function are given by<ref>{{cite journal |url = http://campus.unibo.it/85943/1/mcddmsTranWIR2003.pdf |last1= Chiani|first1= M.|last2= Dardari|first2= D. |last3=Simon |first3= M.K.|date = 2003 |title = New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels|journal = IEEE Transactions on Wireless Communications|volume = 2|number=4|pages = 840–845| doi=10.1109/TWC.2003.814350 | citeseerx= 10.1.1.190.6761}}</ref>
 <math display="block">\begin{align}
-   \operatorname{erfc} x &\leq \frac{1}{2}e^{-2 x^2} + \frac{1}{2}e^{- x^2} \leq e^{-x^2}, &\quad x &> 0 \\[1.5ex]
+   \operatorname{erfc}(x) &\leq \frac{1}{2}e^{-2 x^2} + \frac{1}{2}e^{- x^2} \leq e^{-x^2}, &\quad x &> 0 \\[1.5ex]
-   \operatorname{erfc} x &\approx \frac{1}{6}e^{-x^2} + \frac{1}{2}e^{-\frac{4}{3} x^2}, &\quad x &> 0 .
+   \operatorname{erfc}(x) &\approx \frac{1}{6}e^{-x^2} + \frac{1}{2}e^{-\frac{4}{3} x^2}, &\quad x &> 0 .
 \end{align}</math>
 </li>
 <li>
-The above have been generalized to sums of {{mvar|N}} exponentials<ref>{{cite journal |doi=10.1109/TCOMM.2020.3006902 |title=Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials|journal=IEEE Transactions on Communications |year=2020 |last1=Tanash |first1=I.M. |last2=Riihonen |first2=T. |volume=68 |issue=10 |pages=6514–6524 |arxiv=2007.06939 |s2cid=220514754}}</ref> with increasing accuracy in terms of {{mvar|N}} so that {{math|erfc ''x''}} can be accurately approximated or bounded by {{math|2''Q̃''({{sqrt|2}}''x'')}}, where
+The above have been generalized to sums of {{mvar|N}} exponentials<ref>{{cite journal |doi=10.1109/TCOMM.2020.3006902 |title=Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials|journal=IEEE Transactions on Communications |year=2020 |last1=Tanash |first1=I.M. |last2=Riihonen |first2=T. |volume=68 |issue=10 |pages=6514–6524 |arxiv=2007.06939 |s2cid=220514754}}</ref> with increasing accuracy in terms of {{mvar|N}} so that {{math|erfc(''x'')}} can be accurately approximated or bounded by {{math|2''Q̃''({{sqrt|2}}''x'')}}, where
 <math display="block">\tilde{Q}(x) = \sum_{n=1}^N a_n e^{-b_n x^2}.</math>
 In particular, there is a systematic methodology to solve the numerical coefficients {{math|{(''a<sub>n</sub>'',''b<sub>n</sub>'')}{{su|b=''n'' {{=}} 1|p=''N''}}}} that yield a [[minimax approximation algorithm|minimax]] approximation or bound for the closely related [[Q-function]]: {{math|''Q''(''x'') ≈ ''Q̃''(''x'')}}, {{math|''Q''(''x'') ≤ ''Q̃''(''x'')}}, or {{math|''Q''(''x'') ≥ ''Q̃''(''x'')}} for {{math|''x'' ≥ 0}}. The coefficients {{math|{(''a<sub>n</sub>'',''b<sub>n</sub>'')}{{su|b=''n'' {{=}} 1|p=''N''}}}} for many variations of the exponential approximations and bounds up to {{math|''N'' {{=}} 25}} have been released to open access as a comprehensive dataset.<ref>{{cite journal | doi=10.5281/zenodo.4112978 | title=Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set] | url=https://zenodo.org/record/4112978 | website=Zenodo | year=2020 | last1=Tanash | first1=I.M. | last2=Riihonen | first2=T.}}</ref></li>
@@ Line 257: / Line 257: @@
 <li>
 A tight approximation of the complementary error function for {{math|''x'' ∈ [0,∞)}} is given by [[George Karagiannidis|Karagiannidis]] & Lioumpas (2007)<ref>{{cite journal|last1=Karagiannidis |first1=G. K. |last2=Lioumpas |first2=A. S. |url=http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf |title=An improved approximation for the Gaussian Q-function |date=2007 |journal=IEEE Communications Letters |volume=11 |issue=8 |pages=644–646|doi=10.1109/LCOMM.2007.070470 |s2cid=4043576 }}</ref> who showed for the appropriate choice of parameters {{math|{''A'',''B''}<nowiki/>}} that
-<math display="block">\operatorname{erfc} x \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x}.</math>
+<math display="block">\operatorname{erfc}(x) \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x}.</math>
 They determined {{math|{''A'',''B''} {{=}} {1.98,1.135}<nowiki/>}}, which gave a good approximation for all {{math|''x'' ≥ 0}}. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.<ref>{{cite journal |doi=10.1109/LCOMM.2021.3052257|title=Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function|journal=IEEE Communications Letters | year=2021 | last1=Tanash | first1=I.M.|last2=Riihonen|first2=T.|volume=25|issue=5|pages=1468–1471|arxiv=2101.07631|s2cid=231639206}}</ref>
 </li>
@@ Line 263: / Line 263: @@
 <li>
 A single-term lower bound is<ref>{{cite journal |last1=Chang |first1=Seok-Ho |last2=Cosman |first2=Pamela C. |author-link2 = Pamela Cosman |last3=Milstein |first3=Laurence B. |date=November 2011 |title=Chernoff-Type Bounds for the Gaussian Error Function |url=http://escholarship.org/uc/item/6hw4v7pg |journal=IEEE Transactions on Communications |volume=59 |issue=11 |pages=2939–2944 |doi=10.1109/TCOMM.2011.072011.100049 |s2cid=13636638}}</ref>
-<math display="block" display="block">\operatorname{erfc} x \geq \sqrt{\frac{2 e}{\pi}} \frac{\sqrt{\beta - 1}}{\beta} e^{- \beta x^2}, \qquad x \ge 0,\quad \beta > 1,</math>
+<math display="block" display="block">\operatorname{erfc}(x) \geq \sqrt{\frac{2 e}{\pi}} \frac{\sqrt{\beta - 1}}{\beta} e^{- \beta x^2}, \qquad x \ge 0,\quad \beta > 1,</math>
 where the parameter {{mvar|β}} can be picked to minimize error on the desired interval of approximation.
 </li>
@@ Line 269: / Line 269: @@
 <li>
 Another approximation is given by Sergei Winitzki using his "global Padé approximations":<ref>{{cite book |last=Winitzki |first=Sergei |title=Computational Science and Its Applications – ICCSA 2003 |date=2003  |volume=2667 |chapter=Uniform approximations for transcendental functions |publisher=Springer, Berlin |pages=[https://archive.org/details/computationalsci0000iccs_a2w6/page/780 780–789] |isbn=978-3-540-40155-1 |doi=10.1007/3-540-44839-X_82 |chapter-url-access=registration |chapter-url=https://archive.org/details/computationalsci0000iccs_a2w6 |series=Lecture Notes in Computer Science }}</ref><ref>{{cite journal|last1=Zeng |first1=Caibin |last2=Chen |first2=Yang Cuan |title=Global Padé approximations of the generalized Mittag-Leffler function and its inverse |journal=Fractional Calculus and Applied Analysis |date=2015 |volume=18 |issue=6 | pages=1492–1506 |doi= 10.1515/fca-2015-0086 |quote=Indeed, Winitzki [32] provided the so-called global Padé approximation | arxiv=1310.5592 |s2cid=118148950 }}</ref>{{rp|2–3}}
-<math display="block">\operatorname{erf} x \approx \sgn x \cdot \sqrt{1 - \exp\left(-x^2\frac{\frac{4}{\pi} + ax^2}{1 + ax^2}\right)}</math>
+<math display="block">\operatorname{erf}(x) \approx \sgn x \cdot \sqrt{1 - \exp\left(-x^2\frac{\frac{4}{\pi} + ax^2}{1 + ax^2}\right)}</math>
 where
 <math display="block">a = \frac{8(\pi - 3)}{3\pi(4 - \pi)} \approx 0.140012.</math>
@@ Line 275: / Line 275: @@
 {{pb}}
 This approximation can be inverted to obtain an approximation for the inverse error function:
-<math display="block">\operatorname{erf}^{-1}x \approx \sgn x \cdot \sqrt{\sqrt{\left(\frac{2}{\pi a} + \frac{\ln\left(1 - x^2\right)}{2}\right)^2 - \frac{\ln\left(1 - x^2\right)}{a}} -\left(\frac{2}{\pi a} + \frac{\ln\left(1 - x^2\right)}{2}\right)}.</math>
+<math display="block">\operatorname{erf}^{-1}(x) \approx \sgn x \cdot \sqrt{\sqrt{\left(\frac{2}{\pi a} + \frac{\ln\left(1 - x^2\right)}{2}\right)^2 - \frac{\ln\left(1 - x^2\right)}{a}} -\left(\frac{2}{\pi a} + \frac{\ln\left(1 - x^2\right)}{2}\right)}.</math>
 </li>
 <li>
 An approximation with a maximal error of {{val|1.2e-7}} for any real argument is:<ref>{{cite book | last = Press | first = William H. | title = Numerical Recipes in Fortran 77: The Art of Scientific Computing | isbn = 0-521-43064-X | year = 1992 | page = 214 | publisher = Cambridge University Press }}</ref>
-<math display="block">\operatorname{erf} x = \begin{cases}
+<math display="block">\operatorname{erf}(x) = \begin{cases}
 -\tau &  x\ge 0\\
 \tau-1 &  x < 0
@@ Line 325: / Line 325: @@
 {{further|Interval estimation|Coverage probability|68–95–99.7 rule}}
 {| class="wikitable" style="text-align:left;margin-left:24pt"
-! {{math|''x''}}!! {{math|erf ''x''}} !! {{math|1 − erf ''x''}}
+! {{math|''x''}}!! {{math|erf(''x'')}} !! {{math|1 − erf(''x'')}}
 |-
 |0 || {{val|0}}  || {{val|1}}
@@ Line 396: / Line 396: @@
 ===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]]
+[[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
+\operatorname{erfc}(x)
-& = 1-\operatorname{erf} x \\[5pt]
+& = 1-\operatorname{erf}(x) \\[5pt]
 & = \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\,\mathrm  dt \\[5pt]
-& = e^{-x^2} \operatorname{erfcx} x,
+& = 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>
+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]]
+[[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
+\operatorname{erfi}(x)
-& = -i\operatorname{erf} ix \\[5pt]
+& = -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),
@@ Line 420: / Line 420: @@
 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.
+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]]:
@@ Line 431: / Line 431: @@
 \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} \left(1+\operatorname{erf}\left(\frac{x}{\sqrt 2}\right)\right)\\[6pt]
 &= \frac{1}{2} \operatorname{erfc}\left(-\frac{x}{\sqrt 2}\right)
 \end{align}</math>
@@ Line 443: / Line 443: @@
 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}\\
+Q(x) &= \frac{1}{2} - \frac{1}{2} \operatorname{erf}\left(\frac{x}{\sqrt 2}\right)\\
-&= \frac{1}{2}\operatorname{erfc}\frac{x}{\sqrt 2}.
+&= \frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt 2}\right).
 \end{align}</math>
@@ Line 453: / Line 453: @@
 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>
+<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
+<math display="block">\operatorname{erf}(x)
-= \sgn x \cdot P\left(\tfrac{1}{2}, x^2\right)
+= \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]].
+= \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^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^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^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) \\
+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>
+<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>
+<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>
+<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 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>

Error function: Difference between revisions

Latest revision as of 13:25, 22 June 2025

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Complementary error function

Imaginary error function

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