Gauss error function

Error function
Plot of the error function
General information
General definition
Fields of application	Probability, thermodynamics
Domain, Codomain and Image
Domain
Image
Basic features
Parity	Odd
Specific features
Root	0
Derivative
Antiderivative
Series definition
Taylor series

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as:^[1]

This integral is a special (non-elementary) sigmoid function that occurs often in probability, 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 x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/√2, erf x is the probability that Y falls in the range [−x, x].

Two closely related functions are the complementary error function (erfc) defined as

and the imaginary error function (erfi) defined as

where i is the imaginary unit

Name[edit]

The name «error function» and its abbreviation erf 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.»^[2] The error function complement was also discussed by Glaisher in a separate publication in the same year.^[3]
For the «law of facility» of errors whose density is given by

(the normal distribution), Glaisher calculates the probability of an error lying between p and q as:

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[edit]

When the results of a series of measurements are described by a normal distribution with standard deviation σ and expected value 0, then erf (a/σ √2) is the probability that the error of a single measurement lies between −a and +a, for positive a. 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 X ~ Norm[μ,σ] (a normal distribution with mean μ and standard deviation σ) and a constant L < μ:

where A and B are certain numeric constants. If L is sufficiently far from the mean, specifically μ − L ≥ σ√ln k, then:

so the probability goes to 0 as k → ∞.

The probability for X being in the interval [L_a, L_b] can be derived as

Properties[edit]

The property erf (−z) = −erf z means that the error function is an odd function. This directly results from the fact that the integrand e^−t2 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 z:

where z is the complex conjugate of z.

The integrand f = exp(−z²) and f = erf z are shown in the complex z-plane in the figures at right with domain coloring.

The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf z approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series[edit]

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges, but is famously known «[…] for its bad convergence if x > 1.»^[4]

The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e^−z2 into its Maclaurin series and integrating term by term, one obtains the error function’s Maclaurin series as:

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

because −(2k − 1)z²/k(2k + 1) expresses the multiplier to turn the kth term into the (k + 1)th term (considering z as the first term).

The imaginary error function has a very similar Maclaurin series, which is:

which holds for every complex number z.

Derivative and integral[edit]

The derivative of the error function follows immediately from its definition:

From this, the derivative of the imaginary error function is also immediate:

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

Higher order derivatives are given by

where H are the physicists’ Hermite polynomials.^[5]

Bürmann series[edit]

An expansion,^[6] which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using Hans Heinrich Bürmann’s theorem:^[7]

where sgn is the sign function. By keeping only the first two coefficients and choosing c₁ = 31/200 and c₂ = −341/8000, the resulting approximation shows its largest relative error at x = ±1.3796, where it is less than 0.0036127:

Inverse functions[edit]

Given a complex number z, there is not a unique complex number w satisfying erf w = z, so a true inverse function would be multivalued. However, for −1 < x < 1, there is a unique real number denoted erf⁻¹ x satisfying

The inverse error function is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series

where c₀ = 1 and

So we have the series expansion (common factors have been canceled from numerators and denominators):

(After cancellation the numerator/denominator fractions are entries OEIS: A092676/OEIS: A092677 in the OEIS; without cancellation the numerator terms are given in entry OEIS: A002067.) The error function’s value at ±∞ is equal to ±1.

For |z| < 1, we have erf(erf⁻¹ z) = z.

The inverse complementary error function is defined as

For real x, there is a unique real number erfi⁻¹ x satisfying erfi(erfi⁻¹ x) = x. The inverse imaginary error function is defined as erfi⁻¹ x.^[8]

For any real x, Newton’s method can be used to compute erfi⁻¹ x, and for −1 ≤ x ≤ 1, the following Maclaurin series converges:

where c_k is defined as above.

Asymptotic expansion[edit]

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real x is

where (2n − 1)!! is the double factorial of (2n − 1), which is the product of all odd numbers up to (2n − 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that for any integer N ≥ 1 one has

where the remainder, in Landau notation, is

as x → ∞.

Indeed, the exact value of the remainder is

which follows easily by induction, writing

and integrating by parts.

For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc x (while for not too large values of x, the above Taylor expansion at 0 provides a very fast convergence).

Continued fraction expansion[edit]

A continued fraction expansion of the complementary error function is:^[9]

Integral of error function with Gaussian density function[edit]

which appears related to Ng and Geller, formula 13 in section 4.3^[10] with a change of variables.

Factorial series[edit]

The inverse factorial series:

converges for Re(z²) > 0. Here

zⁿ denotes the rising factorial, and s(n,k) denotes a signed Stirling number of the first kind.^[11][12]
There also exists a representation by an infinite sum containing the double factorial:

Numerical approximations[edit]

Approximation with elementary functions[edit]

• 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:

  

  (maximum error: 5×10⁻⁴)

  where a₁ = 0.278393, a₂ = 0.230389, a₃ = 0.000972, a₄ = 0.078108

  

  (maximum error: 2.5×10⁻⁵)

  where p = 0.47047, a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

  

  (maximum error: 3×10⁻⁷)

  where a₁ = 0.0705230784, a₂ = 0.0422820123, a₃ = 0.0092705272, a₄ = 0.0001520143, a₅ = 0.0002765672, a₆ = 0.0000430638

  

  (maximum error: 1.5×10⁻⁷)

  where p = 0.3275911, a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 1.061405429

  All of these approximations are valid for x ≥ 0. To use these approximations for negative x, use the fact that erf x is an odd function, so erf x = −erf(−x).

• Exponential bounds and a pure exponential approximation for the complementary error function are given by^[13]

  

• The above have been generalized to sums of N exponentials^[14] with increasing accuracy in terms of N so that erfc x can be accurately approximated or bounded by 2Q̃(√2x), where

  

  In particular, there is a systematic methodology to solve the numerical coefficients {(a_n,b_n)}^N
  _{n = 1} that yield a minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(a_n,b_n)}^N
  _{n = 1} for many variations of the exponential approximations and bounds up to N = 25 have been released to open access as a comprehensive dataset.^[15]

• A tight approximation of the complementary error function for x ∈ [0,∞) is given by Karagiannidis & Lioumpas (2007)^[16] who showed for the appropriate choice of parameters {A,B} that

  

  They determined {A,B} = {1.98,1.135}, which gave a good approximation for all x ≥ 0. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.^[17]

• A single-term lower bound is^[18]

  

  where the parameter β can be picked to minimize error on the desired interval of approximation.

• Another approximation is given by Sergei Winitzki using his «global Padé approximations»:^{[19][20]: 2–3}

  

  where

  

  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 x. Using the alternate value a ≈ 0.147 reduces the maximum relative error to about 0.00013.^[21]

  This approximation can be inverted to obtain an approximation for the inverse error function:

  

• An approximation with a maximal error of 1.2×10⁻⁷ for any real argument is:^[22]

  

  with

  

  and

  

Table of values[edit]

[edit]

Complementary error function[edit]

The complementary error function, denoted erfc, 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

which also defines erfcx, the scaled complementary error function^[23] (which can be used instead of erfc to avoid arithmetic underflow^[23][24]). Another form of erfc x for x ≥ 0 is known as Craig’s formula, after its discoverer:^[25]

This expression is valid only for positive values of x, but it can be used in conjunction with erfc x = 2 − erfc(−x) to obtain 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 erfc of the sum of two non-negative variables is as follows:^[26]

Imaginary error function[edit]

The imaginary error function, denoted erfi, is defined as

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow^[23]).

Despite the name «imaginary error function», erfi x is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

Cumulative distribution function[edit]

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by some software languages^{[citation needed]}, as they differ only by scaling and translation. Indeed,

the normal cumulative distribution function plotted in the complex plane

or rearranged for erf and erfc:

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 Φ 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.^{[further explanation needed]}

In terms of the regularized gamma function P and the incomplete gamma function,

sgn x is the sign function.

Generalized error functions[edit]

Some authors discuss the more general functions:^{[citation needed]}

Notable cases are:

• E₀(x) is a straight line through the origin: E₀(x) = x/e√π
• E₂(x) is the error function, erf x.

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the gamma function and incomplete gamma function:

Therefore, we can define the error function in terms of the incomplete gamma function:

Iterated integrals of the complementary error function[edit]

The iterated integrals of the complementary error function are defined by^[27]

The general recurrence formula is

They have the power series

from which follow the symmetry properties

and

Implementations[edit]

As real function of a real argument[edit]

• 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.^[28]
• The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.^[29]

As complex function of a complex argument[edit]

• 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

See also[edit]

[edit]

• Gaussian integral, over the whole real line
• Gaussian function, derivative
• Dawson function, renormalized imaginary error function
• Goodwin–Staton integral

In probability[edit]

• Normal distribution
• Normal cumulative distribution function, a scaled and shifted form of error function
• Probit, the inverse or quantile function of the normal CDF
• Q-function, the tail probability of the normal distribution

References[edit]

Further reading[edit]

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. «Chapter 7». Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 297. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), «Section 6.2. Incomplete Gamma Function and Error Function», Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
• Temme, Nico M. (2010), «Error Functions, Dawson’s and Fresnel Integrals», in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248

External links[edit]

• A Table of Integrals of the Error Functions

Template:Infobox mathematical function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as:^[1]

[math]displaystyle{ operatorname{erf} z = frac{2}{sqrtpi}int_0^z e^{-t^2},dt. }[/math]

This integral is a special (non-elementary) sigmoid function that occurs often in probability, 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 x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/√2, erf x is the probability that Y falls in the range [−x, x].

Two closely related functions are the complementary error function (erfc) defined as

[math]displaystyle{ operatorname{erfc} z = 1 — operatorname{erf} z, }[/math]

and the imaginary error function (erfi) defined as

[math]displaystyle{ operatorname{erfi} z = -ioperatorname{erf} iz, }[/math]

where i is the imaginary unit

Name

The name «error function» and its abbreviation erf 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.»^[2] The error function complement was also discussed by Glaisher in a separate publication in the same year.^[3]
For the «law of facility» of errors whose density is given by

[math]displaystyle{ f(x)=left(frac{c}{pi}right)^frac12 e^{-cx^2} }[/math]

(the normal distribution), Glaisher calculates the probability of an error lying between p and q as:

[math]displaystyle{ left(frac{c}{pi}right)^frac12 int_p^qe^{-cx^2},dx = tfrac12left(operatorname{erf} left(qsqrt{c}right) -operatorname{erf} left(psqrt{c}right)right). }[/math]

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 σ and expected value 0, then erf (a/σ √2) is the probability that the error of a single measurement lies between −a and +a, for positive a. 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 X ~ Norm[μ,σ] (a normal distribution with mean μ and standard deviation σ) and a constant L < μ:

[math]displaystyle{ begin{align}
Pr[Xleq L] &= frac12 + frac12operatorname{erf}frac{L-mu}{sqrt{2}sigma} \
&approx A exp left(-B left(frac{L-mu}{sigma}right)^2right) end{align} }[/math]

where A and B are certain numeric constants. If L is sufficiently far from the mean, specifically μ − L ≥ σ√ln k, then:

[math]displaystyle{ Pr[Xleq L] leq A exp (-B ln{k}) = frac{A}{k^B} }[/math]

so the probability goes to 0 as k → ∞.

The probability for X being in the interval [L_a, L_b] can be derived as

[math]displaystyle{ begin{align}
Pr[L_aleq X leq L_b] &= int_{L_a}^{L_b} frac{1}{sqrt{2pi}sigma} expleft(-frac{(x-mu)^2}{2sigma^2}right) ,dx \
&= frac12left(operatorname{erf}frac{L_b-mu}{sqrt{2}sigma} — operatorname{erf}frac{L_a-mu}{sqrt{2}sigma}right).end{align} }[/math]

Properties

The property erf (−z) = −erf z means that the error function is an odd function. This directly results from the fact that the integrand e^−t2 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 z:

[math]displaystyle{ operatorname{erf} overline{z} = overline{operatorname{erf} z} }[/math]

where z is the complex conjugate of z.

The integrand f = exp(−z²) and f = erf z are shown in the complex z-plane in the figures at right with domain coloring.

The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf z approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges, but is famously known «[…] for its bad convergence if x > 1.»^[4]

The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e^−z2 into its Maclaurin series and integrating term by term, one obtains the error function’s Maclaurin series as:

[math]displaystyle{ begin{align} operatorname{erf} z
&= frac{2}{sqrtpi}sum_{n=0}^inftyfrac{(-1)^n z^{2n+1}}{n! (2n+1)} \[6pt]
&=frac{2}{sqrtpi} left(z-frac{z^3}{3}+frac{z^5}{10}-frac{z^7}{42}+frac{z^9}{216}-cdotsright)
end{align} }[/math]

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

[math]displaystyle{ begin{align} operatorname{erf} z
&= frac{2}{sqrtpi}sum_{n=0}^inftyleft(z prod_{k=1}^n {frac{-(2k-1) z^2}{k (2k+1)}}right) \[6pt]
&= frac{2}{sqrtpi} sum_{n=0}^infty frac{z}{2n+1} prod_{k=1}^n frac{-z^2}{k}
end{align} }[/math]

because −(2k − 1)z²/k(2k + 1) expresses the multiplier to turn the kth term into the (k + 1)th term (considering z as the first term).

The imaginary error function has a very similar Maclaurin series, which is:

[math]displaystyle{ begin{align} operatorname{erfi} z &= frac{2}{sqrtpi}sum_{n=0}^inftyfrac{z^{2n+1}}{n! (2n+1)} \[6pt] &=frac{2}{sqrtpi} left(z+frac{z^3}{3}+frac{z^5}{10}+frac{z^7}{42}+frac{z^9}{216}+cdotsright) end{align} }[/math]

which holds for every complex number z.

Derivative and integral

The derivative of the error function follows immediately from its definition:

[math]displaystyle{ frac{d}{dz}operatorname{erf} z =frac{2}{sqrtpi} e^{-z^2}. }[/math]

From this, the derivative of the imaginary error function is also immediate:

[math]displaystyle{ frac{d}{dz}operatorname{erfi} z =frac{2}{sqrtpi} e^{z^2}. }[/math]

An antiderivative of the error function, obtainable by integration by parts, is

[math]displaystyle{ zoperatorname{erf}z + frac{e^{-z^2}}{sqrtpi}. }[/math]

An antiderivative of the imaginary error function, also obtainable by integration by parts, is

[math]displaystyle{ zoperatorname{erfi}z — frac{e^{z^2}}{sqrtpi}. }[/math]

Higher order derivatives are given by

[math]displaystyle{ operatorname{erf}^{(k)}z = frac{2 (-1)^{k-1}}{sqrtpi} mathit{H}_{k-1}(z) e^{-z^2} = frac{2}{sqrtpi} frac{d^{k-1}}{dz^{k-1}} left(e^{-z^2}right),qquad k=1, 2, dots }[/math]

where H are the physicists’ Hermite polynomials.^[5]

Bürmann series

An expansion,^[6] which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using Hans Heinrich Bürmann’s theorem:^[7]

[math]displaystyle{ begin{align}
operatorname{erf} x &= frac 2 {sqrtpi} 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}{sqrtpi} sgn x cdot sqrt{1-e^{-x^2}} left(frac{sqrtpi}{2} + sum_{k=1}^infty c_k e^{-kx^2} right).
end{align} }[/math]

where sgn is the sign function. By keeping only the first two coefficients and choosing c₁ = 31/200 and c₂ = −341/8000, the resulting approximation shows its largest relative error at x = ±1.3796, where it is less than 0.0036127:

[math]displaystyle{ operatorname{erf} x approx frac{2}{sqrtpi}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

Given a complex number z, there is not a unique complex number w satisfying erf w = z, so a true inverse function would be multivalued. However, for −1 < x < 1, there is a unique real number denoted erf⁻¹ x satisfying

[math]displaystyle{ operatorname{erf}left(operatorname{erf}^{-1} xright) = x. }[/math]

The inverse error function is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series

[math]displaystyle{ operatorname{erf}^{-1} z=sum_{k=0}^inftyfrac{c_k}{2k+1}left (frac{sqrtpi}{2}zright )^{2k+1}, }[/math]

where c₀ = 1 and

[math]displaystyle{ begin{align}
c_k&=sum_{m=0}^{k-1}frac{c_m c_{k-1-m}}{(m+1)(2m+1)} \ &= left{1,1,frac{7}{6},frac{127}{90},frac{4369}{2520},frac{34807}{16200},ldotsright}.
end{align} }[/math]

So we have the series expansion (common factors have been canceled from numerators and denominators):

[math]displaystyle{ operatorname{erf}^{-1} z=frac{sqrt{pi}}{2} left (z+frac{pi}{12}z^3+frac{7pi^2}{480}z^5+frac{127pi^3}{40320}z^7+frac{4369pi^4}{5806080}z^9+frac{34807pi^5}{182476800}z^{11}+cdotsright ). }[/math]

(After cancellation the numerator/denominator fractions are entries OEIS: A092676/OEIS: A092677 in the OEIS; without cancellation the numerator terms are given in entry OEIS: A002067.) The error function’s value at ±∞ is equal to ±1.

For |z| < 1, we have erf(erf⁻¹ z) = z.

The inverse complementary error function is defined as

[math]displaystyle{ operatorname{erfc}^{-1}(1-z) = operatorname{erf}^{-1} z. }[/math]

For real x, there is a unique real number erfi⁻¹ x satisfying erfi(erfi⁻¹ x) = x. The inverse imaginary error function is defined as erfi⁻¹ x.^[8]

For any real x, Newton’s method can be used to compute erfi⁻¹ x, and for −1 ≤ x ≤ 1, the following Maclaurin series converges:

[math]displaystyle{ operatorname{erfi}^{-1} z =sum_{k=0}^inftyfrac{(-1)^k c_k}{2k+1}left (frac{sqrtpi}{2}zright )^{2k+1}, }[/math]

where c_k 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 x is

[math]displaystyle{ begin{align}
operatorname{erfc} x &= frac{e^{-x^2}}{xsqrt{pi}}left(1 + sum_{n=1}^infty (-1)^n frac{1cdot3cdot5cdots(2n — 1)}{left(2x^2right)^n}right) \[6pt]
&= frac{e^{-x^2}}{xsqrt{pi}}sum_{n=0}^infty (-1)^n frac{(2n — 1)!!}{left(2x^2right)^n},
end{align} }[/math]

where (2n − 1)!! is the double factorial of (2n − 1), which is the product of all odd numbers up to (2n − 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that for any integer N ≥ 1 one has

[math]displaystyle{ operatorname{erfc} x = frac{e^{-x^2}}{xsqrt{pi}}sum_{n=0}^{N-1} (-1)^n frac{(2n — 1)!!}{left(2x^2right)^n} + R_N(x) }[/math]

where the remainder, in Landau notation, is

[math]displaystyle{ R_N(x) = Oleft(x^{- (1 + 2N)} e^{-x^2}right) }[/math]

as x → ∞.

Indeed, the exact value of the remainder is

[math]displaystyle{ R_N(x) := frac{(-1)^N}{sqrtpi} 2^{1 — 2N}frac{(2N)!}{N!}int_x^infty t^{-2N}e^{-t^2},dt, }[/math]

which follows easily by induction, writing

[math]displaystyle{ e^{-t^2} = -(2t)^{-1}left(e^{-t^2}right)’ }[/math]

and integrating by parts.

For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc x (while for not too large values of x, the above Taylor expansion at 0 provides a very fast convergence).

Continued fraction expansion

A continued fraction expansion of the complementary error function is:^[9]

[math]displaystyle{ operatorname{erfc} z = frac{z}{sqrtpi}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]

Integral of error function with Gaussian density function

[math]displaystyle{ int_{-infty}^{infty} operatorname{erf} left(ax+b right) frac{1}{sqrt{2pi sigma^2}}expleft(-frac{(x-mu)^2}{2 sigma^2} right),dx= operatorname{erf} frac{amu+b}{sqrt{1+2 a^2 sigma^2}} , qquad a,b,mu,sigma in R }[/math]

which appears related to Ng and Geller, formula 13 in section 4.3^[10] with a change of variables.

Factorial series

The inverse factorial series:

[math]displaystyle{ begin{align}
operatorname{erfc} z &= frac{e^{-z^2}}{sqrt{pi},z} sum_{n=0}^infty frac{(-1)^n Q_n}{{(z^2+1)}^{bar{n}}}\
&= frac{e^{-z^2}}{sqrt{pi},z}left(1 -frac12frac{1}{(z^2+1)} + frac{1}{4}frac{1}{(z^2+1)(z^2+2)} — cdots right)
end{align} }[/math]

converges for Re(z²) > 0. Here

[math]displaystyle{ begin{align} Q_n
&overset{text{def}}{{}={}} frac{1}{Gammaleft(frac12right)} int_0^infty tau(tau-1)cdots(tau-n+1)tau^{-frac12} e^{-tau} ,dtau \
&= sum_{k=0}^n left(tfrac12right)^{bar{k}} s(n,k), end{align} }[/math]

zⁿ denotes the rising factorial, and s(n,k) denotes a signed Stirling number of the first kind.^[11][12]
There also exists a representation by an infinite sum containing the double factorial:

[math]displaystyle{ operatorname{erf} z = frac{2}{sqrtpi} sum_{n=0}^infty frac{(-2)^n(2n-1)!!}{(2n+1)!}z^{2n+1} }[/math]

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:

  [math]displaystyle{ operatorname{erf} x approx 1 — frac{1}{left(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4right)^4}, qquad x geq 0 }[/math]

  (maximum error: 5×10⁻⁴)

  where a₁ = 0.278393, a₂ = 0.230389, a₃ = 0.000972, a₄ = 0.078108

  [math]displaystyle{ operatorname{erf} x approx 1 — left(a_1t + a_2t^2 + a_3t^3right)e^{-x^2},quad t=frac{1}{1 + px}, qquad x geq 0 }[/math]

  (maximum error: 2.5×10⁻⁵)

  where p = 0.47047, a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

  [math]displaystyle{ operatorname{erf} x approx 1 — frac{1}{left(1 + a_1x + a_2x^2 + cdots + a_6x^6right)^{16}}, qquad x geq 0 }[/math]
  (maximum error: 3×10⁻⁷)

  where a₁ = 0.0705230784, a₂ = 0.0422820123, a₃ = 0.0092705272, a₄ = 0.0001520143, a₅ = 0.0002765672, a₆ = 0.0000430638

  [math]displaystyle{ operatorname{erf} x approx 1 — left(a_1t + a_2t^2 + cdots + a_5t^5right)e^{-x^2},quad t = frac{1}{1 + px} }[/math]
  (maximum error: 1.5×10⁻⁷)

  where p = 0.3275911, a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 1.061405429

  All of these approximations are valid for x ≥ 0. To use these approximations for negative x, use the fact that erf x is an odd function, so erf x = −erf(−x).

• Exponential bounds and a pure exponential approximation for the complementary error function are given by^[13]

  [math]displaystyle{ begin{align}
  operatorname{erfc} x &leq tfrac12e^{-2 x^2} + tfrac12e^{- x^2} leq e^{-x^2}, &quad x &gt 0 \
  operatorname{erfc} x &approx tfrac16e^{-x^2} + tfrac12e^{-frac43 x^2}, &quad x &gt 0 .
  end{align} }[/math]

• The above have been generalized to sums of N exponentials^[14] with increasing accuracy in terms of N so that erfc x can be accurately approximated or bounded by 2Q̃(√2x), where

  [math]displaystyle{ 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 {(a_n,b_n)}Nn = 1 that yield a minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(a_n,b_n)}Nn = 1 for many variations of the exponential approximations and bounds up to N = 25 have been released to open access as a comprehensive dataset.^[15]

• A tight approximation of the complementary error function for x ∈ [0,∞) is given by Karagiannidis & Lioumpas (2007)^[16] who showed for the appropriate choice of parameters {A,B} that

  [math]displaystyle{ operatorname{erfc} x approx frac{left(1 — e^{-Ax}right)e^{-x^2}}{Bsqrt{pi} x}. }[/math]

  They determined {A,B} = {1.98,1.135}, which gave a good approximation for all x ≥ 0. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.^[17]

• A single-term lower bound is^[18]

  [math]displaystyle{ operatorname{erfc} x geq sqrt{frac{2 e}{pi}} frac{sqrt{beta — 1}}{beta} e^{- beta x^2}, qquad x ge 0,quad beta gt 1, }[/math]

  where the parameter β can be picked to minimize error on the desired interval of approximation.

• Another approximation is given by Sergei Winitzki using his «global Padé approximations»:^{[19][20]:2–3}

  [math]displaystyle{ operatorname{erf} x approx sgn x cdot sqrt{1 — expleft(-x^2frac{frac{4}{pi} + ax^2}{1 + ax^2}right)} }[/math]

  where

  [math]displaystyle{ a = frac{8(pi — 3)}{3pi(4 — pi)} approx 0.140012. }[/math]

  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 x. Using the alternate value a ≈ 0.147 reduces the maximum relative error to about 0.00013.^[21]

  This approximation can be inverted to obtain an approximation for the inverse error function:

  [math]displaystyle{ operatorname{erf}^{-1}x approx sgn x cdot sqrt{sqrt{left(frac{2}{pi a} + frac{lnleft(1 — x^2right)}{2}right)^2 — frac{lnleft(1 — x^2right)}{a}} -left(frac{2}{pi a} + frac{lnleft(1 — x^2right)}{2}right)}. }[/math]

• An approximation with a maximal error of 1.2×10⁻⁷ for any real argument is:^[22]

  [math]displaystyle{ operatorname{erf} x = begin{cases}
  1-tau & xge 0\
  tau-1 & x lt 0
  end{cases} }[/math]

  with

  [math]displaystyle{ begin{align}
  tau &= tcdotexpleft(-x^2-1.26551223+1.00002368 t+0.37409196 t^2+0.09678418 t^3 -0.18628806 t^4right.\
  &left. qquadqquadqquad +0.27886807 t^5-1.13520398 t^6+1.48851587 t^7 -0.82215223 t^8+0.17087277 t^9right)
  end{align} }[/math]

  and

  [math]displaystyle{ t=frac{1}{1+frac12|x|}. }[/math]

Table of values

Complementary error function

The complementary error function, denoted erfc, 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

[math]displaystyle{ begin{align}
operatorname{erfc} x
& = 1-operatorname{erf} x \[5pt]
& = frac{2}{sqrtpi} int_x^infty e^{-t^2}, dt \[5pt]
& = e^{-x^2} operatorname{erfcx} x,
end{align} }[/math]

which also defines erfcx, the scaled complementary error function^[23] (which can be used instead of erfc to avoid arithmetic underflow^[23][24]). Another form of erfc x for x ≥ 0 is known as Craig’s formula, after its discoverer:^[25]

[math]displaystyle{ operatorname{erfc} (x mid xge 0)
= frac 2 pi int_0^frac{pi}{2} exp left( — frac{x^2}{sin^2 theta} right) , dtheta. }[/math]

This expression is valid only for positive values of x, but it can be used in conjunction with erfc x = 2 − erfc(−x) to obtain 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 erfc of the sum of two non-negative variables is as follows:^[26]

[math]displaystyle{ operatorname{erfc} (x+y mid x,yge 0) = frac 2 pi int_0^frac{pi}{2} exp left( — frac{x^2}{sin^2 theta} — frac{y^2}{cos^2 theta} right) , dtheta. }[/math]

Imaginary error function

The imaginary error function, denoted erfi, is defined as

[math]displaystyle{ begin{align}
operatorname{erfi} x
& = -ioperatorname{erf} ix \[5pt]
& = frac{2}{sqrtpi} int_0^x e^{t^2},dt \[5pt]
& = frac{2}{sqrtpi} e^{x^2} D(x),
end{align} }[/math]

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow^[23]).

Despite the name «imaginary error function», erfi x is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

[math]displaystyle{ w(z) = e^{-z^2}operatorname{erfc}(-iz) = operatorname{erfcx}(-iz). }[/math]

Cumulative distribution function

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

[math]displaystyle{ begin{align}Phi(x)
&=frac{1}{sqrt{2pi}}int_{-infty}^x e^tfrac{-t^2}{2},dt\[6pt]
&= frac12 left(1+operatorname{erf}frac{x}{sqrt 2}right)\[6pt]
&=frac12operatorname{erfc}left(-frac{x}{sqrt 2}right)
end{align} }[/math]

or rearranged for erf and erfc:

[math]displaystyle{ begin{align}
operatorname{erf}(x) &= 2 Phi left ( x sqrt{2} right ) — 1 \[6pt]
operatorname{erfc}(x) &= 2 Phi left ( — x sqrt{2} right ) \ &=2left(1-Phi left ( x sqrt{2} right)right).
end{align} }[/math]

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]displaystyle{ begin{align}
Q(x) &=frac12 — frac12 operatorname{erf} frac{x}{sqrt 2}\
&=frac12operatorname{erfc}frac{x}{sqrt 2}.
end{align} }[/math]

The inverse of Φ is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

[math]displaystyle{ operatorname{probit}(p) = Phi^{-1}(p) = sqrt{2}operatorname{erf}^{-1}(2p-1) = -sqrt{2}operatorname{erfc}^{-1}(2p). }[/math]

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):

[math]displaystyle{ operatorname{erf} x = frac{2x}{sqrtpi} Mleft(tfrac12,tfrac32,-x^2right). }[/math]

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

In terms of the regularized gamma function P and the incomplete gamma function,

[math]displaystyle{ operatorname{erf} x = sgn x cdot Pleft(tfrac12, x^2right) = frac{sgn x}{sqrtpi}gammaleft(tfrac12, x^2right). }[/math]

sgn x is the sign function.

Generalized error functions

Some authors discuss the more general functions:

[math]displaystyle{ E_n(x) = frac{n!}{sqrtpi} int_0^x e^{-t^n},dt =frac{n!}{sqrtpi}sum_{p=0}^infty(-1)^pfrac{x^{np+1}}{(np+1)p!}.
}[/math]

Notable cases are:

• E₀(x) is a straight line through the origin: E₀(x) = x/e√π
• E₂(x) is the error function, erf x.

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the gamma function and incomplete gamma function:

[math]displaystyle{ E_n(x) = frac{1}{sqrtpi}Gamma(n)left(Gammaleft(frac{1}{n}right) — Gammaleft(frac{1}{n}, x^nright)right), qquad xgt 0. }[/math]

Therefore, we can define the error function in terms of the incomplete gamma function:

[math]displaystyle{ operatorname{erf} x = 1 — frac{1}{sqrtpi}Gammaleft(tfrac12, x^2right). }[/math]

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by^[27]

[math]displaystyle{ begin{align}
operatorname{i}^n!operatorname{erfc} z &= int_z^infty operatorname{i}^{n-1}!operatorname{erfc} zeta, dzeta \[6pt]
operatorname{i}^0!operatorname{erfc} z &= operatorname{erfc} z \
operatorname{i}^1!operatorname{erfc} z &= operatorname{ierfc} z = frac{1}{sqrtpi} e^{-z^2} — z operatorname{erfc} z \
operatorname{i}^2!operatorname{erfc} z &= tfrac14 left( operatorname{erfc} z -2 z operatorname{ierfc} z right) \
end{align} }[/math]

The general recurrence formula is

[math]displaystyle{ 2 n cdot operatorname{i}^n!operatorname{erfc} z = operatorname{i}^{n-2}!operatorname{erfc} z -2 z cdot operatorname{i}^{n-1}!operatorname{erfc} z }[/math]

They have the power series

[math]displaystyle{ operatorname{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]displaystyle{ operatorname{i}^{2m}!operatorname{erfc} (-z) =-operatorname{i}^{2m}!operatorname{erfc} z +sum_{q=0}^m frac{z^{2q}}{2^{2(m-q)-1}(2q)! (m-q)!} }[/math]

and

[math]displaystyle{ operatorname{i}^{2m+1}!operatorname{erfc}(-z) =operatorname{i}^{2m+1}!operatorname{erfc} z +sum_{q=0}^m frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)! (m-q)!}.
}[/math]

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.^[28]
• The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.^[29]

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

See also

• Gaussian integral, over the whole real line
• Gaussian function, derivative
• Dawson function, renormalized imaginary error function
• Goodwin–Staton integral

In probability

• Normal distribution
• Normal cumulative distribution function, a scaled and shifted form of error function
• Probit, the inverse or quantile function of the normal CDF
• Q-function, the tail probability of the normal distribution

References

Further reading

• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). «Chapter 7». Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 297. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_297.htm.
• Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), «Section 6.2. Incomplete Gamma Function and Error Function», Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, http://apps.nrbook.com/empanel/index.html#pg=259
• Temme, Nico M. (2010), «Error Functions, Dawson’s and Fresnel Integrals», in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/7

External links

• MathWorld – Erf
• A Table of Integrals of the Error Functions

{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B=

}}

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:^[1][2]

The complementary error function, denoted erfc, is defined as

which also defines erfcx, the scaled complementary error function^[3] (which can be used instead of erfc to avoid arithmetic underflow^[3][4]).

The imaginary error function, denoted erfi, is defined as

,

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow^[3]).

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

The name «error function»

The error function is used in measurement theory (using probability and statistics), and although its use in other branches of mathematics has nothing to do with the characterization of measurement errors, the name has stuck.

The error function is related to the cumulative distribution , the integral of the standard normal distribution, by^[2]

The error function, evaluated at for positive x values, gives the probability that a measurement, under the influence of normally distributed errors with standard deviation , has a distance less than x from the mean value.^[5] This function is used in statistics to predict behavior of any sample with respect to the population mean. This usage is similar to the Q-function, which in fact can be written in terms of the error function.

Properties

{{#invoke:Multiple image|render}}

The property means that the error function is an odd function. This directly results from the fact that the integrand is an even function.

For any complex number z:

where is the complex conjugate of z.

The integrand ƒ = exp(−z²) and ƒ = erf(z) are shown in the complex z-plane in figures 2 and 3. Level of Im(ƒ) = 0 is shown with a thick green line. Negative integer values of Im(ƒ) are shown with thick red lines. Positive integer values of Im(f) are shown with thick blue lines. Intermediate levels of Im(ƒ) = constant are shown with thin green lines. Intermediate levels of Re(ƒ) = constant are shown with thin red lines for negative values and with thin blue lines for positive values.

At the real axis, erf(z) approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.

The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e^−z2 into its Taylor series and integrating term by term, one obtains the error function’s Taylor series as:

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

because expresses the multiplier to turn the k^th term into the (k + 1)^th term (considering z as the first term).

The error function at +∞ is exactly 1 (see Gaussian integral).

The derivative of the error function follows immediately from its definition:

An antiderivative of the error function is

Bürmann series

An expansion,^[6] which converges more rapidly for all real values of than a Taylor expansion, is obtained by using Heinrich H. Bürmann´s theorem:^[7]

By keeping only the first two coefficients and choosing and , the resulting approximation shows its largest relative error at , where it is less than  :

Inverse functions

The inverse error function can be defined in terms of the Maclaurin series

where c₀ = 1 and

So we have the series expansion (note that common factors have been canceled from numerators and denominators):

(After cancellation the numerator/denominator fractions are entries A092676/A132467 in the OEIS; without cancellation the numerator terms are given in entry A002067.) Note that the error function’s value at ±∞ is equal to ±1.

The inverse complementary error function is defined as

Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real x is

where (2n – 1)!! is the double factorial: the product of all odd numbers up to (2n – 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that, for any one has

where the remainder, in Landau notation, is

as .

Indeed, the exact value of the remainder is

which follows easily by induction, writing and integrating by parts.

For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc(x) (while for not too large values of x note that the above Taylor expansion at 0 provides a very fast convergence).

Continued fraction expansion

A continued fraction expansion of the complementary error function is:^[8]

Integral of error function with Gaussian density function

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:

    (maximum error: 5×10⁻⁴)

where a₁ = 0.278393, a₂ = 0.230389, a₃ = 0.000972, a₄ = 0.078108

    (maximum error: 2.5×10⁻⁵)

where p = 0.47047, a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

    (maximum error: 3×10⁻⁷)

where a₁ = 0.0705230784, a₂ = 0.0422820123, a₃ = 0.0092705272, a₄ = 0.0001520143, a₅ = 0.0002765672, a₆ = 0.0000430638

    (maximum error: 1.5×10⁻⁷)

where p = 0.3275911, a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 1.061405429

All of these approximations are valid for x ≥ 0. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x) = −erf(−x).

Another approximation is given by

where

This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the error is less than 0.00035 for all x. Using the alternate value a ≈ 0.147 reduces the maximum error to about 0.00012.^[9]

This approximation can also be inverted to calculate the inverse error function:

Exponential bounds and a pure exponential approximation for the complementary error function are given by ^[10]

A single-term lower bound is
^[11]

where the parameter β can be picked to minimize error on the desired interval of approximation.

Numerical approximation

Over the complete range of values, there is an approximation with a maximal error of , as follows:^[12]

with

and

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 −a and +a, for positive a. 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 is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by software languages, as they differ only by scaling and translation. Indeed,

or rearranged for erf and erfc:

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.Template:Elucidate

In terms of the Regularized Gamma function P and the incomplete gamma function,

is the sign function.

Generalized error functions

Some authors discuss the more general functions:{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]}
}}

Notable cases are:

• E₀(x) is a straight line through the origin:
• E₂(x) is the error function, erf(x).

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the Gamma function and incomplete Gamma function:

Therefore, we can define the error function in terms of the incomplete Gamma function:

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by

They have the power series

from which follow the symmetry properties

and

Implementations

• C: C99 provides the functions double erf(double x) and double erfc(double x) in the header math.h. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. For complex double arguments, the function names cerf and cerfc are «reserved for future use»; the missing implementation is provided by the open-source project libcerf, which is based on the Faddeeva package.

• C++: C++11 provides erf() and erfc() in the header cmath. Both functions are overloaded to accept arguments of type float, double, and long double. For complex<double>, the Faddeeva package provides a C++ complex<double> implementation.

• Excel: Microsoft Excel provides the erf, and the erfc functions, nonetheless both inverse functions are not in the current library.^[13]

• Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED functions to calculate the error function and its complement for real arguments. Fortran 77 implementations are available in SLATEC.

• Google search: Google’s search also acts as a calculator and will evaluate «erf(…)» and «erfc(…)» for real arguments.

• Haskell: An erf package^[14] exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

• IDL: provides both erf and erfc for real and complex arguments.

• Java: Apache commons-math^[15] provides implementations of erf and erfc for real arguments.

• Maple: Maple implements both erf and erfc for real and complex arguments.

• MathCAD provides both erf(x) and erfc(x) for real arguments.

• Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha.

• Matlab provides both erf and erfc for real arguments, also via W. J. Cody’s algorithm.^[16]

• Maxima provides both erf and erfc for real and complex arguments.

• PARI/GP: provides erfc for real and complex arguments, via tanh-sinh quadrature plus special cases.

• Perl: erf (for real arguments, using Cody’s algorithm^[16]) is implemented in the Perl module Math::SpecFun

• Python: Included since version 2.7 as math.erf() for real arguments. For previous versions or for complex arguments, SciPy includes implementations of erf, erfc, erfi, and related functions for complex arguments in scipy.special.^[17] A complex-argument erf is also in the arbitrary-precision arithmetic mpmath library as mpmath.erf()

• R: «The so-called ‘error function'»^[18] is not provided directly, but is detailed as an example of the normal cumulative distribution function (?pnorm), which is based on W. J. Cody’s rational Chebyshev approximation algorithm.^[16]

• Ruby: Provides Math.erf() and Math.erfc() for real arguments.

See also

• Gaussian integral, over the whole real line
• Gaussian function, derivative
• Dawson function, renormalized imaginary error function

In probability

• Normal distribution
• Normal cumulative distribution function, a scaled and shifted form of error function
• Probit, the inverse or quantile function of the normal CDF
• Q-function, the tail probability of the normal distribution

References

• Template:AS ref
• {{#invoke:citation/CS1|citation

|CitationClass=citation
}}

• Template:Dlmf

External links

• MathWorld – Erf
• Error-function numerical table and calculator

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:^[1][2]

The complementary error function, denoted erfc, is defined as

which also defines erfcx, the scaled complementary error function^[3] (which can be used instead of erfc to avoid arithmetic underflow^[3][4]). Another form of is known as Craig’s formula:^[5]

The imaginary error function, denoted erfi, is defined as

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow^[3]).

Despite the name «imaginary error function», is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

The name «error function»

The error function is used in measurement theory (using probability and statistics), and its use in other branches of mathematics is typically unrelated to the characterization of measurement errors.

The error function is related to the cumulative distribution , the integral of the standard normal distribution, by^[2]

The error function, evaluated at for positive x values, gives the probability that a measurement, under the influence of normally distributed errors with standard deviation , has a distance less than x from the mean value.^[6] This function is used in statistics to predict behavior of any sample with respect to the population mean. This usage is similar to the Q-function, which in fact can be written in terms of the error function.

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.

For any complex number z:

where is the complex conjugate of z.

The integrand ƒ = exp(−z²) and ƒ = erf(z) are shown in the complex z-plane in figures 2 and 3. Level of Im(ƒ) = 0 is shown with a thick green line. Negative integer values of Im(ƒ) are shown with thick red lines. Positive integer values of Im(f) are shown with thick blue lines. Intermediate levels of Im(ƒ) = constant are shown with thin green lines. Intermediate levels of Re(ƒ) = constant are shown with thin red lines for negative values and with thin blue lines for positive values.

The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf(z) approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.

The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e^−z2 into its Maclaurin series and integrating term by term, one obtains the error function’s Maclaurin series as:

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

because expresses the multiplier to turn the k^th term into the (k + 1)^th term (considering z as the first term).

The imaginary error function has a very similar Maclaurin series, which is:

which holds for every complex number z.

Derivative and integral

The derivative of the error function follows immediately from its definition:

From this, the derivative of the imaginary error function is also immediate:

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

Higher order derivatives are given by

where are the probabilists’ Hermite polynomials.^[7]

Bürmann series

An expansion,^[8] which converges more rapidly for all real values of than a Taylor expansion, is obtained by using Hans Heinrich Bürmann’s theorem:^[9]

By keeping only the first two coefficients and choosing and , the resulting approximation shows its largest relative error at , where it is less than  :

Inverse functions

Given complex number z, there is not a unique complex number w satisfying , so a true inverse function would be multivalued. However, for -1<x<1, there is a unique real number denoted satisfying .

The inverse error function is usually defined with domain (-1,1), and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk |z|<1 of the complex plane, using the Maclaurin series

where c₀ = 1 and

So we have the series expansion (note that common factors have been canceled from numerators and denominators):

(After cancellation the numerator/denominator fractions are entries A092676/A132467 in the OEIS; without cancellation the numerator terms are given in entry A002067.) Note that the error function’s value at ±∞ is equal to ±1.

For |z|<1, we have .

The inverse complementary error function is defined as

For real x, there is a unique real number satisfying . The inverse imaginary error function is defined as .^[10]

For any real x, Newton’s method can be used to compute , and for , the following Maclaurin series converges:

where c_k 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 x is

where (2n – 1)!! is the double factorial: the product of all odd numbers up to (2n – 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that, for any one has

where the remainder, in Landau notation, is

as

Indeed, the exact value of the remainder is

which follows easily by induction, writing and integrating by parts.

For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc(x) (while for not too large values of x note that the above Taylor expansion at 0 provides a very fast convergence).

Continued fraction expansion

A continued fraction expansion of the complementary error function is:^[11]

Integral of error function with Gaussian density function

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:

    (maximum error: 5×10⁻⁴)

where a₁ = 0.278393, a₂ = 0.230389, a₃ = 0.000972, a₄ = 0.078108

    (maximum error: 2.5×10⁻⁵)

where p = 0.47047, a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

    (maximum error: 3×10⁻⁷)

where a₁ = 0.0705230784, a₂ = 0.0422820123, a₃ = 0.0092705272, a₄ = 0.0001520143, a₅ = 0.0002765672, a₆ = 0.0000430638

    (maximum error: 1.5×10⁻⁷)

where p = 0.3275911, a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 1.061405429

All of these approximations are valid for x ≥ 0. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x) = −erf(−x).

Another approximation is given by

where

This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the error is less than 0.00035 for all x. Using the alternate value a ≈ 0.147 reduces the maximum error to about 0.00012.^[12]

This approximation can also be inverted to calculate the inverse error function:

Exponential bounds and a pure exponential approximation for the complementary error function are given by ^[13]

A single-term lower bound is ^[14]

where the parameter β can be picked to minimize error on the desired interval of approximation.

Numerical approximation

Over the complete range of values, there is an approximation with a maximal error of , as follows:^[15]

with

and

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 −a and +a, for positive a. 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. Given random variable and constant :

where A and B are certain numeric constants. If L is sufficiently far from the mean, i.e. , then:

so the probability goes to 0 as .

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

or rearranged for erf and erfc:

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.^{[further explanation needed]}

In terms of the regularized Gamma function P and the incomplete gamma function,

is the sign function.

Generalized error functions

Some authors discuss the more general functions:^{[citation needed]}

Notable cases are:

• E₀(x) is a straight line through the origin:
• E₂(x) is the error function, erf(x).

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the Gamma function and incomplete Gamma function:

Therefore, we can define the error function in terms of the incomplete Gamma function:

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by

They have the power series

from which follow the symmetry properties

and

Implementations

• C: C99 provides the functions double erf(double x) and double erfc(double x) in the header math.h. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. For complex double arguments, the function names cerf and cerfc are «reserved for future use»; the missing implementation is provided by the open-source project libcerf, which is based on the Faddeeva package.

• C++: C++11 provides erf() and erfc() in the header cmath. Both functions are overloaded to accept arguments of type float, double, and long double. For complex<double>, the Faddeeva package provides a C++ complex<double> implementation.

• Excel: Microsoft Excel provides the erf, and the erfc functions, nonetheless both inverse functions are not in the current library.^[16]

• Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED functions to calculate the error function and its complement for real arguments. Fortran 77 implementations are available in SLATEC.

• Google search: Google’s search also acts as a calculator and will evaluate «erf(…)» and «erfc(…)» for real arguments.

• Haskell: An erf package^[17] exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

• IDL: provides both erf and erfc for real and complex arguments.

• Java: Apache commons-math^[18] provides implementations of erf and erfc for real arguments.

• Julia: Includes erf and erfc for real and complex arguments. Also has erfi for calculating

• Maple: Maple implements both erf and erfc for real and complex arguments.

• MathCAD provides both erf(x) and erfc(x) for real arguments.

• Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha.

• Matlab provides both erf and erfc for real arguments, also via W. J. Cody’s algorithm.^[19]

• Maxima provides both erf and erfc for real and complex arguments.

• PARI/GP: provides erfc for real and complex arguments, via tanh-sinh quadrature plus special cases.

• Perl: erf (for real arguments, using Cody’s algorithm^[19]) is implemented in the Perl module Math::SpecFun

• Python: Included since version 2.7 as math.erf() for real arguments. For previous versions or for complex arguments, SciPy includes implementations of erf, erfc, erfi, and related functions for complex arguments in scipy.special.^[20] A complex-argument erf is also in the arbitrary-precision arithmetic mpmath library as mpmath.erf()

• R: «The so-called ‘error function'»^[21] is not provided directly, but is detailed as an example of the normal cumulative distribution function (?pnorm), which is based on W. J. Cody’s rational Chebyshev approximation algorithm.^[19]

• Ruby: Provides Math.erf() and Math.erfc() for real arguments.

See also

• Gaussian integral, over the whole real line
• Gaussian function, derivative
• Dawson function, renormalized imaginary error function
• Goodwin–Staton integral

In probability

• Normal distribution
• Normal cumulative distribution function, a scaled and shifted form of error function
• Probit, the inverse or quantile function of the normal CDF
• Q-function, the tail probability of the normal distribution

References

• Abramowitz, Milton; Stegun, Irene A., eds. (December 1972) [1964]. «Chapter 7». Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (10 ed.). New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 297. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642.<templatestyles src=»Module:Citation/CS1/styles.css»></templatestyles>
• Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), «Section 6.2. Incomplete Gamma Function and Error Function», Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8<templatestyles src=»Module:Citation/CS1/styles.css»></templatestyles>

• Temme, Nico M. (2010), «Error Functions, Dawson’s and Fresnel Integrals», in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248CS1 maint: display-editors (link)<templatestyles src=»Module:Citation/CS1/styles.css»></templatestyles>

External links

• MathWorld – Erf
• Error-function numerical table and calculator

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File:Error Function.svg

In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. It is defined as:

The complementary error function, denoted erfc, is defined in terms of the error function:

The complex error function, denoted w(x), (also known as the Faddeeva function) is also defined in terms of the error function:

Properties

The error function is odd:

Also, for any complex number x one has

where is the complex conjugate of x.

The integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand in a Taylor series, one obtains the Taylor series for the error function as follows:

which holds for every real number x, and also throughout the complex plane. This result arises from the Taylor series expansion of which is and is then integrated term by term. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternate formulation may be useful:

because expresses the multiplier to turn the i^th term into the (i+1)^th term (assuming we number the «x» as the first term).

The error function at infinity is exactly 1 (see Gaussian integral).

The derivative of the error function follows immediately from its definition:

The inverse error function has series

where c₀ = 1 and

So we have the series expansion (note that common factors have been canceled from numerators and denominators):

[1]

(After cancellation the numerator/denominator fractions are entries A092676/A132467 in the OEIS; without cancellation the numerator terms are given in entry A002067.)

File:Error Function Complementary.svg

Note that error function’s value at plus/minus infinity is equal to plus/minus 1.

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 −a and +a, for positive a.

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.

In digital optical communication system, BER is expressed by:

Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large x is

This series diverges for every finite x. However, in practice only the first few terms of this expansion are needed to obtain a good approximation of erfc(x), whereas the Taylor series given above converges very slowly.

Another approximation is given by

where

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

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 P and the incomplete gamma function,

is the sign function.

Generalised error functions

File:Error Function Generalised.svg

Some authors discuss the more general functions

Notable cases are:

• E₀(x) is a straight line through the origin:
• E₂(x) is the error function, erf(x).

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n>0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x>0 using the Gamma function:

Therefore, we can define the error function in terms of the Gamma function:

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by

They have the power series

from which follow the symmetry properties

and

Implementation

C/C++: It is provided by C99 as the functions double erf(double x) and double erfc(double x) in the header math.h or cmath. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. GCC makes these functions available in C++ too.

Fortran: E.g. gfortran provides the intrinsic real function ERF(X) and the double precision function DERF(X).

See also

• Gaussian function
• Statistical significance

References

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 7)

External links

• MathWorld — Erf