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In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as:
(When x is negative, the integral is interpreted as the negative of the integral from x to zero.)
The complementary error function, denoted erfc, is defined as
The imaginary error function, denoted erfi, is defined as
The complex error function, denoted w(x) and also known as the Faddeeva function, is defined as
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 (the "bell curve"), by
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. 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.
Plots in the complex plane
The property means that the error function is an odd 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 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∞.
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 kth 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
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
A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large 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
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).
A continued fractions expansion of the complementary error function is :
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:
where a1=0.278393, a2=0.230389, a3=0.000972, a4=0.078108
where p=0.47047, a1=0.3480242, a2=-0.0958798, a3=0.7478556
where a1=0.0705230784, a2=0.0422820123, a3=0.0092705272, a4=0.0001520143, a5=0.0002765672, a6=0.0000430638
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.
This approximation can also be inverted to calculate the inverse error function:
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.[further explanation needed]
In terms of the Regularized Gamma function P and the incomplete gamma function,
is the sign function.
Some authors discuss the more general functions:[citation needed]
Notable cases are:
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.
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:
The iterated integrals of the complementary error function are defined by
They have the power series
from which follow the symmetry properties
and
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