In mathematics, the error function (also called the Gauss error function), often denoted by , is a function defined as:
The integral here is a complex contour integral which is path-independent because is holomorphic on the whole complex plane . 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 . 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 , is the probability that falls in the range .
Two closely related functions are the complementary error function erfc: defined as
and the imaginary error function erfi: defined as
where is the imaginary unit.
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
(the normal distribution), Glaisher calculates the probability of an error lying between and as
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:
where and are certain numeric constants. If is sufficiently far from the mean, specifically , then:
so the probability goes to 0 as .
The probability for being in the interval can be derived as
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 maps real numbers to real numbers, for any complex number :
where denotes the complex conjugate of .
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 âÂÂ1 at . At the imaginary axis, it tends to .
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 impractical.
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, integrating term by term, and using the fact that , one obtains the error function's Maclaurin series as:
which holds for every complex number . The denominator terms are sequence in the OEIS.
It is a special case of Kummer's function:
For iterative calculation of the above series, the following alternative formulation may be useful:
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:
which holds for every complex number .
The derivative of the error function follows immediately from its definition:
From this, the derivative of the imaginary error function is also immediate: Higher order derivatives are given by
where are the physicists' Hermite polynomials.
An antiderivative of the error function, obtainable by integration by parts, is
An antiderivative of the imaginary error function, also obtainable by integration by parts, is
An expansion which converges more rapidly for all real values of than a Taylor expansion is obtained by using Hans Heinrich Bürmann's theorem:
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:
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
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
where and
So we have the series expansion (common factors have been canceled from numerators and denominators):
(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
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:
where is defined as above.
A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real is
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
where the remainder is
which follows easily by induction, writing
and integrating by parts.
The asymptotic behavior of the remainder term, in Landau notation, is
as . This can be found by
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).
A continued fraction expansion of the complementary error function was found by Laplace:
The inverse factorial series:
converges for . Here
denotes the rising factorial, and denotes a signed Stirling number of the first kind. The Taylor series can be written in terms of the double factorial:
Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25âÂÂ28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are:
(maximum error: )
where , , ,
(maximum error: )
where , , ,
(maximum error: )
where , , , , ,
(maximum error: )
where , , , , ,
One can improve the accuracy of the A&S approximation by extending it with three extra parameters,
where p1 = 0.406742016006509, p2 = 0.0072279182302319, a1 = 0.316879890481381, a2 = -0.138329314150635, a3 = 1.08680830347054, a4 = -1.11694155120396, a5 = 1.20644903073232, a6 = -0.393127715207728, a7 = 0.0382613542530727. The maximum error of this approximation is about . The parameters are obtained by fitting the extended approximation to the accurate values of the error function using the following Python code.
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
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
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
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
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":
where
This is designed to be very accurate in the neighborhoods of 0 and 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.
The extended "global Pade" approximation,
provides a maximum error of about , as demonstrated by the following Python script.
Winitzki's approximation can be inverted to obtain an approximation for the inverse error function:
An approximation with a maximal error of for any real argument is:
An approximation of with a maximum relative error less than in absolute value is: for
and for
A simple approximation for real-valued arguments can be done through hyperbolic functions:
which keeps the absolute difference
Since the error function and the Gaussian Q-function are closely related through the identity or equivalently , 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 was introduced by Abreu (2012) based on a simple algebraic expression with only two exponential terms:
These bounds stem from a unified form where the parameters and are selected to ensure the bounding properties: for the lower bound, and , and for the upper bound, and . 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 for positive integers via the binomial theorem, suggesting potential adaptability for powers of , though this is less commonly required in error function applications.
The complementary error function, denoted , is defined as
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:
This expression is valid only for positive values of , but 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
The imaginary error function, denoted , is defined as
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:
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,
or rearranged for and :
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
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
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):
It has a simple expression in terms of the Fresnel integral.
In terms of the regularized gamma function and the incomplete gamma function, is the sign function.
The iterated integrals of the complementary error function are defined by
The general recurrence formula is
They have the power series
from which follow the symmetry properties
and