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:
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 variance 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.
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." The error function complement was also discussed by Glaisher in a separate publication in the same year. For the "law of facility" of errors whose density is given by
(the normal distribution), Glaisher calculates the chance of an error lying between and as:
When the results of a series of measurements are described by a normal distribution with standard deviation and expected value 0, then is the probability that the error of a single measurement lies between −a and +a, for positive a. This is useful, for example, in determining the bit error rate of a digital communication system.
The error function and its approximations can be used to estimate results that hold with high probability or with low 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 .
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∞.
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."
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:
For iterative calculation of the above series, the following alternative formulation may be useful:
because expresses the multiplier to turn the kth term into the (k + 1)st 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 integralEdit
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 imaginary error function, also obtainable by integration by parts, is
Higher order derivatives are given by
( 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 :
Given a 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 (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 .
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 .
For any real x, Newton's method can be used to compute , and for , the following Maclaurin series converges:
where ck is defined as above.
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
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 expansionEdit
Integral of error function with Gaussian density functionEdit
which appears related to Ng and Geller, formula 13 in section 4.3 with a change of variables.
- The inverse factorial series:
- converges for Here
- denotes the rising factorial, and denotes a signed Stirling number of the first kind.
- Representation by an infinite sum containing the double factorial:
Approximation with elementary functionsEdit
- 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 
- The above have been generalized to sums of exponentials with increasing accuracy in terms of so that can be accurately approximated or bounded by , where
- In particular, there is a systematic methodology to solve the numerical coefficients that yield a minimax approximation or bound for the closely related Q-function: , , or for . The coefficients for many variations of the exponential approximations and bounds up to have been released to open access as a comprehensive dataset.
- A tight approximation of the complementary error function for is given by Karagiannidis & Lioumpas (2007) who showed for the appropriate choice of parameters that
- They determined which gave a good approximation for all . Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.
- A single-term lower bound is
- where the parameter 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"::2–3
- 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.
- This approximation can be inverted to obtain an approximation for the inverse error function:
An approximation with a maximal error of for any real argument is:
Table of valuesEdit
Complementary error functionEdit
The complementary error function, denoted , is defined as
which also defines , the scaled complementary error function (which can be used instead of erfc to avoid arithmetic underflow). Another form of for non-negative is known as Craig's formula, after its discoverer:
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 of the sum of two non-negative variables is as follows:
Imaginary error functionEdit
The imaginary error function, denoted erfi, is defined as
Despite the name "imaginary error function", is real when x is real.
Cumulative distribution functionEdit
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,
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 standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.
is the sign function.
Generalized error functionsEdit
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 En for odd n look similar (but not identical) to each other. Similarly, the En 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.
Therefore, we can define the error function in terms of the incomplete Gamma function:
Iterated integrals of the complementary error functionEdit
The iterated integrals of the complementary error function are defined by
The general recurrence formula is
They have the power series
from which follow the symmetry properties
As real function of a real argumentEdit
- 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 erfc, erfcl.
- The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.
As complex function of a complex argumentEdit
- Gaussian integral, over the whole real line
- Gaussian function, derivative
- Dawson function, renormalized imaginary error function
- Goodwin–Staton integral
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