1. Name

2. Applications

3. Properties
    Taylor series  Derivative and integral  Bürmann series  Inverse functions  Asymptotic expansion  Continued fraction expansion  Integral of error function with Gaussian density function  Factorial series 

4. Numerical approximations
    Approximation with elementary functions  Polynomial  Table of values 

5. Related functions
    Complementary error function  Imaginary error function  Cumulative distribution function  Generalized error functions  Iterated integrals of the complementary error function 

6. See also
     Related functions   In probability 

7. References

8. Further reading

9. External links

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]

In statistics, for nonnegative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and variance 0.5, erf(x) describes the probability of Y falling in the range [−x, x].

There are several closely related functions, such as the complementary error function, the imaginary error function, and others.

Name

The name "error function" and its abbreviation {{math|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."^[3] The error function complement was also discussed by Glaisher in a separate publication in the same year.^[4]

For the "law of facility" of errors whose density is given by (the normal distribution), Glaisher calculates the chance of an error lying between and as:

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 .

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 f = exp(−z²) and f = erf(z) are shown in the complex z-plane in figures 2 and 3. Level of Im(f) = 0 is shown with a thick green line. Negative integer values of Im(f) are shown with thick red lines. Positive integer values of Im(f) are shown with thick blue lines. Intermediate levels of Im(f) = constant are shown with thin green lines. Intermediate levels of Re(f) = 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 physicists' Hermite polynomials.^[5]

Bürmann series

An expansion,^[6] which converges more rapidly for all real values of than a Taylor expansion, is obtained by using Hans Heinrich 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

Given complex number z, there is not a unique complex number w satisfying , so a true inverse function would be multivalued. However, for {{math|−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 {{math|{{!}}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 {{oeis|A092676}}/{{oeis|A092677}} in the OEIS; without cancellation the numerator terms are given in entry {{oeis|A002067}}.) Note that the error function's value at ±∞ is equal to ±1.

For {{math|{{!}}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 .^[8]

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 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 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:^[9]

Integral of error function with Gaussian density function

Factorial series

The inverse factorial series

converges for Here

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:

    (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 ^[12]

• A tight approximation of the complementary error function for is given by Karagiannidis & Lioumpas (2007)^[13] who showed for the appropriate choice of parameters that

They determined which gave a good approximation for all

• A single-term lower bound is^[14]

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":^{[15] [16]}{{rp|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.^[17]

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

Polynomial

An approximation with a maximal error of for any real argument is:^[18]

with

and

Table of values

Related functions

Complementary error function

The complementary error function, denoted , is defined as

which also defines , the scaled complementary error function^[19] (which can be used instead of erfc to avoid arithmetic underflow^[19][20]). Another form of for non-negative is known as Craig’s formula, after its discoverer:^[21]

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.

Imaginary error function

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^[19]).

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:

Cumulative distribution 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.{{Elucidate|date=May 2012}}

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|date=August 2011}}

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^[22]

The general recurrence formula is

They have the power series

from which follow the symmetry properties

and

See also

Related functions

• 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

• {{AS ref |7|297}}
• {{Citation |last1=Press |first1=William H. |last2=Teukolsky |first2=Saul A. |last3=Vetterling |first3=William T. |last4=Flannery |first4=Brian P. |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=Cambridge University Press |location=New York |isbn=978-0-521-88068-8 |chapter=Section 6.2. Incomplete Gamma Function and Error Function |chapter-url=http://apps.nrbook.com/empanel/index.html#pg=259 }}
• {{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=Nico M. |last=Temme }}

External links

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

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