Error Function
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In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a
special Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer Literature * ''Specia ...
(non-
elementary Elementary may refer to: Arts, entertainment, and media Music * ''Elementary'' (Cindy Morgan album), 2001 * ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, a ...
)
sigmoid Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include: * Sigmoid function, a mathematical function * Sigmoid colon, part of the l ...
function that occurs often in
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of , the error function has the following interpretation: for a random variable that is normally distributed with
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
0 and standard deviation , is the probability that falls in the range . Two closely related functions are the complementary error function () defined as :\operatorname z = 1 - \operatorname z, and the imaginary error function () defined as :\operatorname z = -i\operatorname iz, where is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...


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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
is given by :f(x)=\left(\frac\right)^\frac12 e^ (the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
), Glaisher calculates the probability of an error lying between and as: :\left(\frac\right)^\frac12 \int_p^qe^\,\mathrm dx = \tfrac12\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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
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 In digital transmission, the number of bit errors is the number of received bits of a data stream over a communication channel that have been altered due to noise, interference, distortion or bit synchronization errors. The bit error rate (BER) ...
of a digital communication system. The error and complementary error functions occur, for example, in solutions of the heat equation when
boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s are given by the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
. The error function and its approximations can be used to estimate results that hold
with high probability In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number ''n'' and goes to 1 as ''n'' goes to infinity, i.e. the probability of the event occurring can be ma ...
or with low probability. Given a random variable (a normal distribution with mean and standard deviation ) and a constant : :\begin \Pr \leq L&= \frac12 + \frac12\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 \\ &= \frac12\left(\operatorname\frac - \operatorname\frac\right).\end


Properties

The property means that the error function is an
odd function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
. 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 In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
which takes real numbers to real numbers, for any
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
: :\operatorname \overline = \overline where is the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of ''z''. The integrand and are shown in the complex -plane in the figures at right with
domain coloring In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, d ...
. The error function at is exactly 1 (see
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
). 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 In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
; it has no singularities (except that at infinity) and its
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
always converges, but is famously known " ..for its bad convergence if ." The defining integral cannot be evaluated in closed form in terms of
elementary functions In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
, but by expanding the
integrand In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
into its
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
 . The denominator terms are sequence A007680 in the
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
. 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
 .


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 In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
of the error function, obtainable by
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
, is :z\operatornamez + \frac. An antiderivative of the imaginary error function, also obtainable by integration by parts, is :z\operatornamez - \frac. 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 Hans Heinrich Bürmann (died 21 June 1817, in Mannheim) was a German mathematician and teacher. He ran an "academy of commerce" in Mannheim since 1795 where he used to teach mathematics. He also served as a censor in Mannheim. He was nominated He ...
's theorem: : \begin \operatorname x &= \frac 2 \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 PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&= \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 0.0036127: : \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 \\ &= \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+\fracz^9+\fracz^+\cdots\right ). (After cancellation the numerator/denominator fractions are entries / in the
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
; without cancellation the numerator terms are given in entry .) 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 (\fracz\right )^, where is defined as above.


Asymptotic expansion

A useful
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
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, in
Landau notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
, is :R_N(x) = O\left(x^ e^\right) as . Indeed, the exact value of the remainder is :R_N(x) := \frac 2^\frac\int_x^\infty t^e^\,\mathrm dt, which follows easily by induction, writing :e^ = -(2t)^\left(e^\right)' and integrating by parts. 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 In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
expansion of the complementary error function is: :\operatorname z = \frace^ \cfrac,\qquad a_m = \frac.


Integral of error function with Gaussian density function

:\int_^ \operatorname \left(ax+b \right) \frac\exp\left(-\frac \right)\,\mathrm dx= \operatorname \frac , \qquad a,b,\mu,\sigma \in \R which appears related to Ng and Geller, formula 13 in section 4.3 with a change of variables.


Factorial series

The inverse
factorial series In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...
: :\begin \operatorname z &= \frac \sum_^\infty \frac\\ &= \frac\left(1 -\frac12\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 \\ &= \sum_^n \left(\tfrac12\right)^ s(n,k), \end denotes the
rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
, 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^


Numerical approximations


Approximation with elementary functions


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 The term arithmetic underflow (also floating point underflow, or just underflow) is a condition in a computer program where the result of a calculation is a number of more precise absolute value than the computer can actually represent in memory ...
). Another form of for is known as Craig's formula, after its discoverer: :\operatorname (x \mid x\ge 0) = \frac 2 \pi \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 2 \pi \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 In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function. Definition The Dawson function is defined as either: D_+(x) = e^ \int_0^x e^\,dt, als ...
(which can be used instead of to avoid
arithmetic overflow Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
). Despite the name "imaginary error function", is real when is real. When the error function is evaluated for arbitrary
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 &= \frac12 \left(1+\operatorname\frac\right)\\ pt&=\frac12\operatorname\left(-\frac\right) \end or rearranged for and : :\begin \operatorname(x) &= 2 \Phi \left ( x \sqrt \right ) - 1 \\ pt \operatorname(x) &= 2 \Phi \left ( - x \sqrt \right ) \\ &=2\left(1-\Phi \left ( x \sqrt \right)\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) &=\frac12 - \frac12 \operatorname \frac\\ &=\frac12\operatorname\frac. \end The inverse of is known as the normal quantile function, or
probit In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and s ...
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 In mathematics, the Mittag-Leffler function E_ is a special function, a complex function which depends on two complex parameters \alpha and \beta. It may be defined by the following series when the real part of \alpha is strictly positive: :E_ ...
, and can also be expressed as a
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
(Kummer's function): :\operatorname x = \frac M\left(\tfrac12,\tfrac32,-x^2\right). It has a simple expression in terms of the
Fresnel integral 250px, Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below). The Fresnel integrals and are two transcendental functions n ...
. In terms of the
regularized gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...
and the incomplete gamma function, :\operatorname x = \sgn x \cdot P\left(\tfrac12, x^2\right) = \frac\gamma\left(\tfrac12, x^2\right). is the sign function.


Generalized error functions

Some authors discuss the more general functions: :E_n(x) = \frac \int_0^x e^\,\mathrm dt =\frac\sum_^\infty(-1)^p\frac. Notable cases are: * is a straight line through the origin: * is the error function, . After division by , all the for odd look similar (but not identical) to each other. Similarly, the for even look similar (but not identical) to each other after a simple division by . All generalised error functions for look similar on the positive side of the graph. These generalised functions can equivalently be expressed for using the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and incomplete gamma function: :E_n(x) = \frac\Gamma(n)\left(\Gamma\left(\frac\right) - \Gamma\left(\frac, x^n\right)\right), \qquad x>0. Therefore, we can define the error function in terms of the incomplete gamma function: :\operatorname x = 1 - \frac\Gamma\left(\tfrac12, x^2\right).


Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by :\begin \operatorname^n\!\operatorname z &= \int_z^\infty \operatorname^\!\operatorname \zeta\,\mathrm d\zeta \\ pt\operatorname^0\!\operatorname z &= \operatorname z \\ \operatorname^1\!\operatorname z &= \operatorname z = \frac e^ - z \operatorname z \\ \operatorname^2\!\operatorname z &= \tfrac14 \left( \operatorname z -2 z \operatorname z \right) \\ \end The general recurrence formula is :2 n \cdot \operatorname^n\!\operatorname z = \operatorname^\!\operatorname z -2 z \cdot \operatorname^\!\operatorname z They have the power series :\operatorname^n\!\operatorname z =\sum_^\infty \frac, from which follow the symmetry properties :\operatorname^\!\operatorname (-z) =-\operatorname^\!\operatorname z +\sum_^m \frac and :\operatorname^\!\operatorname(-z) =\operatorname^\!\operatorname z +\sum_^m \frac.


Implementations


As real function of a real argument

* In
Posix The Portable Operating System Interface (POSIX) is a family of standards specified by the IEEE Computer Society for maintaining compatibility between operating systems. POSIX defines both the system- and user-level application programming inter ...
-compliant operating systems, the header
math.h C mathematical operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. All functions use floating-point numbers in one manner or another. Different C standards provid ...
shall declare and the mathematical library
libm C mathematical operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. All functions use floating-point numbers in one manner or another. Different C standards provide d ...
shall provide the functions erf and erfc (
double precision Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Flo ...
) as well as their
single precision Single-precision floating-point format (sometimes called FP32 or float32) is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. A floatin ...
and extended precision counterparts erff, erfl and erfcf, erfcl. * The
GNU Scientific Library The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C; wrappers are available for other programming languages. The GSL is part of the GNU Project and is d ...
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


See also


Related functions

*
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
, over the whole real line *
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is ...
, derivative *
Dawson function In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function. Definition The Dawson function is defined as either: D_+(x) = e^ \int_0^x e^\,dt, als ...
, renormalized imaginary error function *
Goodwin–Staton integral In mathematics the Goodwin–Staton integral is defined as : Frank William John Olver (ed.), N. M. Temme (Chapter contr.), NIST Handbook of Mathematical Functions, Chapter 7, p160,Cambridge University Press 2010 : G(z)=\int_0^\infty \frac \, dt ...


In probability

*
Normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
* Normal cumulative distribution function, a scaled and shifted form of error function *
Probit In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and s ...
, the inverse or quantile function of the normal CDF * Q-function, the tail probability of the normal distribution


References


Further reading

* * *


External links


A Table of Integrals of the Error Functions
{{Authority control Special hypergeometric functions Gaussian function Functions related to probability distributions Analytic functions