error function

In mathematics, the error function (also called the Gauss error function), often denoted by , is a function $\mathrm: \mathbb \to \mathbb$ defined as:
$\operatorname z = \frac\int_0^z e^\,\mathrm dt.$

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

In some old texts,
the error function is defined without the factor of $\frac$.
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 , the error function has the following interpretation: for a real random variable that is normally distributed with mean 0 and standard deviation $\frac$, is the probability that falls in the range .

Two closely related functions are the complementary error function $\mathrm: \mathbb \to \mathbb$ is defined as

$\operatorname z = 1 - \operatorname z,$

and the imaginary error function $\mathrm: \mathbb \to \mathbb$ is defined as

$\operatorname z = -i\operatorname iz,$

where is the imaginary unit.

Name

The name "error function" and its abbreviation 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." The error function complement was also discussed by Glaisher in a separate publication in the same year.
For the "law of facility" of errors whose density is given by
$f(x) = \left(\frac\right)^ e^$
(the normal distribution), Glaisher calculates the probability of an error lying between and as:
$\left(\frac\right)^\frac \int_p^qe^\,\mathrm dx = \tfrac\left(\operatorname \left(q\sqrt\right) -\operatorname \left(p\sqrt\right)\right).$

Applications

When the results of a series of measurements are described by a normal distribution with standard deviation and expected value 0, then is the probability that the error of a single measurement lies between and , for positive . 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 (a normal distribution with mean and standard deviation ) and a constant , it can be shown via integration by substitution:
\begin
\Pr \leq L&= \frac + \frac \operatorname\frac \\
&\approx A \exp \left(-B \left(\frac\right)^2\right)
\end

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

\Pr \leq L\leq A \exp (-B \ln) = \frac

so the probability goes to 0 as .

The probability for being in the interval can be derived as
\begin
\Pr _a\leq X \leq L_b&= \int_^ \frac \exp\left(-\frac\right) \,\mathrm dx \\
&= \frac\left(\operatorname\frac - \operatorname\frac\right).\end

Properties

The property means that the error function is an odd function. This directly results from the fact that the integrand 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 :
$\operatorname \overline = \overline$
where $\overline$ denotes the complex conjugate of $z$.

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

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

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges. For , 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 into its Maclaurin series and integrating term by term, one obtains the error function's Maclaurin series as:
\begin
\operatorname z
&= \frac\sum_^\infty\frac \\ pt&= \frac \left(z-\frac+\frac-\frac+\frac-\cdots\right)
\end
which holds for every complex number . The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:
\begin
\operatorname z
&= \frac\sum_^\infty\left(z \prod_^n \right) \\ pt&= \frac \sum_^\infty \frac \prod_^n \frac
\end
because expresses the multiplier to turn the th term into the th term (considering as the first term).

The imaginary error function has a very similar Maclaurin series, which is:
\begin
\operatorname z
&= \frac\sum_^\infty\frac \\ pt &=\frac \left(z+\frac+\frac+\frac+\frac+\cdots\right)
\end
which holds for every complex number .

Derivative and integral

The derivative of the error function follows immediately from its definition:
$\frac\operatorname z =\frac e^.$
From this, the derivative of the imaginary error function is also immediate:
$\frac\operatorname z =\frac e^.$
An antiderivative of the error function, obtainable by integration by parts, is
$z\operatornamez + \frac+C.$
An antiderivative of the imaginary error function, also obtainable by integration by parts, is
$z\operatornamez - \frac+C.$
Higher order derivatives are given by
$\operatorname^z = \frac \mathit_(z) e^ = \frac \frac \left(e^\right),\qquad k=1, 2, \dots$
where are the physicists' Hermite polynomials.

Bürmann series

An expansion, which converges more rapidly for all real values of than a Taylor expansion, is obtained by using Hans Heinrich Bürmann's theorem:
\begin
\operatorname x
&= \frac \sgn x \cdot \sqrt \left( 1-\frac \left (1-e^ \right ) -\frac \left (1-e^ \right )^2 -\frac \left (1-e^ \right )^3-\frac \left (1-e^ \right )^4 - \cdots \right) \\ 0pt&= \frac \sgn x \cdot \sqrt \left(\frac + \sum_^\infty c_k e^ \right).
\end
where is the sign function. By keeping only the first two coefficients and choosing and , the resulting approximation shows its largest relative error at , where it is less than 0.0034361:
$\operatorname x \approx \frac\sgn x \cdot \sqrt \left(\frac + \frace^-\frac e^\right).$

Inverse functions

Given a complex number , there is not a ''unique'' complex number satisfying , so a true inverse function would be multivalued. However, for , there is a unique ''real'' number denoted satisfying
$\operatorname\left(\operatorname^ x\right) = x.$

The inverse error function is usually defined with domain , and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk of the complex plane, using the Maclaurin series
$\operatorname^ z=\sum_^\infty\frac\left (\fracz\right )^,$
where and
\begin
c_k & =\sum_^\frac \\ ex&= \left\.
\end

So we have the series expansion (common factors have been canceled from numerators and denominators):
$\operatorname^ z = \frac \left (z + \fracz^3 + \fracz^5 + \fracz^7 + \frac z^9 + \fracz^ + \cdots\right ).$
(After cancellation the numerator and denominator values in and respectively; without cancellation the numerator terms are values in .) The error function's value at  is equal to .

For , we have .

The inverse complementary error function is defined as
$\operatorname^(1-z) = \operatorname^ z.$
For real , there is a unique ''real'' number satisfying . The inverse imaginary error function is defined as .

For any real ''x'', Newton's method can be used to compute , and for , the following Maclaurin series converges:
$\operatorname^ z =\sum_^\infty\frac \left( \frac z \right)^,$
where 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 is
\begin
\operatorname x &= \frac\left(1 + \sum_^\infty (-1)^n \frac\right) \\ pt
&= \frac\sum_^\infty (-1)^n \frac,
\end
where is the double factorial of , which is the product of all odd numbers up to . This series diverges for every finite , and its meaning as asymptotic expansion is that for any integer one has
$\operatorname x = \frac\sum_^ (-1)^n \frac + R_N(x)$
where the remainder is
$R_N(x) := \frac \int_x^\infty t^e^\,\mathrm dt,$
which follows easily by induction, writing
$e^ = -\frac \, \frac e^$
and integrating by parts.

The asymptotic behavior of the remainder term, in Landau notation, is
$R_N(x) = O\left(x^ e^\right)$
as . This can be found by
$R_N(x) \propto \int_x^\infty t^e^\,\mathrm dt = e^ \int_0^\infty (t+x)^e^\,\mathrm dt\leq e^ \int_0^\infty x^ e^\,\mathrm dt \propto x^e^.$
For large enough values of , only the first few terms of this asymptotic expansion are needed to obtain a good approximation of (while for not too large values of , 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:
$\operatorname z = \frace^ \cfrac,\qquad a_m = \frac.$

Factorial series

The inverse factorial series:
\begin
\operatorname z
&= \frac \sum_^\infty \frac \\ ex&= \frac \left -\frac\frac + \frac\frac - \cdots \right\end
converges for . Here
\begin
Q_n
&\overset
\frac \int_0^\infty \tau(\tau-1)\cdots(\tau-n+1)\tau^ e^ \,d\tau \\ ex&= \sum_^n \left(\frac\right)^ s(n,k),
\end
denotes the rising factorial, and denotes a signed Stirling number of the first kind.
There also exists a representation by an infinite sum containing the double factorial:
$\operatorname z = \frac \sum_^\infty \fracz^$

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:
  $\operatorname x \approx 1 - \frac, \qquad x \geq 0$
  (maximum error: )

  where , , ,

  $\operatorname x \approx 1 - \left(a_1t + a_2t^2 + a_3t^3\right)e^,\quad t=\frac, \qquad x \geq 0$
  (maximum error: )

  where , , ,

  $\operatorname x \approx 1 - \frac, \qquad x \geq 0$
  (maximum error: )

  where , , , , ,

  $\operatorname x \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5\right)e^,\quad t = \frac$
  (maximum error: )

  where , , , , ,

  All of these approximations are valid for . To use these approximations for negative , use the fact that is an odd function, so .
  

• 
  Exponential bounds and a pure exponential approximation for the complementary error function are given by
  \begin
  \operatorname x &\leq \frace^ + \frace^ \leq e^, &\quad x &> 0 \\ .5ex \operatorname x &\approx \frace^ + \frace^, &\quad x &> 0 .
  \end
  
• 
  The above have been generalized to sums of exponentials with increasing accuracy in terms of so that can be accurately approximated or bounded by , where
  $\tilde(x) = \sum_^N a_n e^.$
  In particular, there is a systematic methodology to solve the numerical coefficients that yield a minimax approximation or bound for the closely related Q-function: , , or for . The coefficients for many variations of the exponential approximations and bounds up to have been released to open access as a comprehensive dataset.
• 
  A tight approximation of the complementary error function for is given by Karagiannidis & Lioumpas (2007) who showed for the appropriate choice of parameters that
  $\operatorname x \approx \frac.$
  They determined , which gave a good approximation for all . Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.
  
• 
  A single-term lower bound is
  $\operatorname x \geq \sqrt \frac e^, \qquad x \ge 0,\quad \beta > 1,$
  where the parameter can be picked to minimize error on the desired interval of approximation.
  
• 
  Another approximation is given by Sergei Winitzki using his "global Padé approximations":
  $\operatorname x \approx \sgn x \cdot \sqrt$
  where
  $a = \frac \approx 0.140012.$
  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 . Using the alternate value reduces the maximum relative error to about 0.00013.

  This approximation can be inverted to obtain an approximation for the inverse error function:
  $\operatorname^x \approx \sgn x \cdot \sqrt.$
  

• 
  An approximation with a maximal error of for any real argument is:
  $\operatorname x = \begin 1-\tau & x\ge 0\\ \tau-1 & x < 0 \end$
  with
  $\begin \tau &= t\cdot\exp\left(-x^2-1.26551223+1.00002368 t+0.37409196 t^2+0.09678418 t^3 -0.18628806 t^4\right.\\ &\left. \qquad\qquad\qquad +0.27886807 t^5-1.13520398 t^6+1.48851587 t^7 -0.82215223 t^8+0.17087277 t^9\right) \end$
  and
  $t = \frac.$
  
• An approximation of $\operatorname$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value is:
  for
  $\begin \operatorname \left(x\right) & = \left(\frac\right) \left(\frac\right) \\ & \left(\frac\right) \left(\frac\right) \\ & \left(\frac\right) \left(\frac\right) e^ \\ \end$
  and for $x<0$
  $\operatorname \left(x\right) = 2 - \operatorname \left(-x\right)$
  


• 
  A simple approximation for real-valued arguments could be done through Hyperbolic functions:
  $\operatorname \left(x\right) \approx z(x) = \tanh\left(\frac\left(x+\fracx^3\right)\right)$
  which keeps the absolute difference
  


• 
  Since the error function and the Gaussian Q-function are closely related through the identity $\operatorname(x) = 2 Q(\sqrt x)$ or equivalently $Q(x) = \frac \operatorname\left(\frac\right)$, 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) based on a simple algebraic expression with only two exponential terms:
  $Q(x) \geq \frac e^ + \frac e^, \qquad x \geq 0,$
  and
  $Q(x) \leq \frac e^ + \frac e^, \qquad x \geq 0.$

  These bounds stem from a unified form $Q_(x; a, b) = \frac + \frac,$ where the parameters $a$ and $b$ are selected to ensure the bounding properties: for the lower bound, $a_ = 12$ and $b_ = \sqrt$, and for the upper bound, $a_ = 50$ and $b_ = 2$.
  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 $\operatorname(x)$, though this is less commonly required in error function applications.
  

Table of values

Related functions

Complementary error function

The complementary error function, denoted , is defined as

\begin
\operatorname x
& = 1-\operatorname x \\ pt& = \frac \int_x^\infty e^\,\mathrm dt \\ pt& = e^ \operatorname x,
\end
which also defines , the scaled complementary error function (which can be used instead of to avoid arithmetic underflow). Another form of for is known as Craig's formula, after its discoverer:
$\operatorname (x \mid x\ge 0) = \frac \int_0^\frac \exp \left( - \frac \right) \, \mathrm d\theta.$
This expression is valid only for positive values of , but it can be used in conjunction with to obtain for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the of the sum of two non-negative variables is as follows:
$\operatorname (x+y \mid x,y\ge 0) = \frac \int_0^\frac \exp \left( - \frac - \frac \right) \,\mathrm d\theta.$

Imaginary error function

The imaginary error function, denoted , is defined as

\begin
\operatorname x
& = -i\operatorname ix \\ pt& = \frac \int_0^x e^\,\mathrm dt \\ pt& = \frac e^ D(x),
\end
where is the Dawson function (which can be used instead of to avoid arithmetic overflow).

Despite the name "imaginary error function", is real when is real.

When the error function is evaluated for arbitrary complex arguments , the resulting complex error function is usually discussed in scaled form as the Faddeeva function:
$w(z) = e^\operatorname(-iz) = \operatorname(-iz).$

Cumulative distribution function

The error function is essentially identical to the standard normal cumulative distribution function, denoted , also named by some software languages, as they differ only by scaling and translation. Indeed,

\begin
\Phi(x)
&= \frac \int_^x e^\tfrac\,\mathrm dt\\ pt
&= \frac \left(1+\operatorname\frac\right)\\ pt&= \frac \operatorname\left(-\frac\right)
\end
or rearranged for and :
\begin
\operatorname(x) &= 2 \Phi - 1 \\ pt \operatorname(x) &= 2 \Phi \\
&= 2\left(1 - \Phi\right).
\end

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
$\begin Q(x) &= \frac - \frac \operatorname \frac\\ &= \frac\operatorname\frac. \end$

The inverse of is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as
$\operatorname(p) = \Phi^(p) = \sqrt\operatorname^(2p-1) = -\sqrt\operatorname^(2p).$

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):
$\operatorname x = \frac M\left(\tfrac,\tfrac,-x^2\right).$

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

In terms of the regularized gamma function and the incomplete gamma function,
$\operatorname x = \sgn x \cdot P\left(\tfrac, x^2\right) = \frac \gamma.$ is the sign function.

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by
\begin
i^n\!\operatorname z &= \int_z^\infty i^\!\operatorname \zeta\,\mathrm d\zeta \\ pti^0\!\operatorname z &= \operatorname z \\
i^1\!\operatorname z &= \operatorname z = \frac e^ - z \operatorname z \\
i^2\!\operatorname z &= \tfrac \left( \operatorname z -2 z \operatorname z \right) \\
\end

The general recurrence formula is
$2 n \cdot i^n\!\operatorname z = i^\!\operatorname z -2 z \cdot i^\!\operatorname z$

They have the power series
$i^n\!\operatorname z =\sum_^\infty \frac,$
from which follow the symmetry properties
$i^\!\operatorname (-z) =-i^\!\operatorname z +\sum_^m \frac$
and
$i^\!\operatorname(-z) =i^\!\operatorname z +\sum_^m \frac.$

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.
* The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.

As complex function of a complex argument

* libcerf
/code>, 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 th
MIT Faddeeva Package

References

Further reading

*
*
*

External links

A Table of Integrals of the Error Functions

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Special hypergeometric functions
Gaussian function
Functions related to probability distributions
Analytic functions