Error Function
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Error Function

Plot of the error 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]

This integral is a special (non-elementary) and 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.

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

(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 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 .

Properties

Plots in the complex plane
Integrand exp(-z2)
erf(z)

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, 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 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 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

Inverse error function

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 / in the OEIS; without cancellation the numerator terms are given in entry .) 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 .[8]

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.

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, 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

converges for Here
   
denotes the rising factorial, and denotes a signed Stirling number of the first kind.[10][11]

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]
  • The above have been generalized to sums of exponentials[13] 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.[14]
  • A tight approximation of the complementary error function for is given by Karagiannidis & Lioumpas (2007)[15] who showed for the appropriate choice of parameters that
They determined which gave a good approximation for all
  • A single-term lower bound is[16]
where the parameter ? can be picked to minimize error on the desired interval of approximation.
  • Another approximation is given by Sergei Winitzki using his "global Padé approximations":[17][18]: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.[19]
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:[20]

with

and

Table of values

x erf(x) 1-erf(x)
0
0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
3
3.5

Related functions

Complementary error function

The complementary error function, denoted , is defined as

which also defines , the scaled complementary error function[21] (which can be used instead of erfc to avoid arithmetic underflow[21][22]). Another form of for non-negative is known as Craig's formula, after its discoverer:[23]

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:[24]

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[21]).

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 ?, 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 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

Graph of generalised error functions En(x):
grey curve: E1(x) = (1 - e -x)/
red curve: E2(x) = erf(x)
green curve: E3(x)
blue curve: E4(x)
gold curve: E5(x).

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.

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[25]

The general recurrence formula is

They have the power series

from which follow the symmetry properties

and

Implementations

As real function of a real argument

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

Related functions

In probability

References

  1. ^ Andrews, Larry C. (1998). Special functions of mathematics for engineers. SPIE Press. p. 110. ISBN 9780819426161.
  2. ^ Glaisher, James Whitbread Lee (July 1871). "On a class of definite integrals". London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 4. 42 (277): 294-302. doi:10.1080/14786447108640568. Retrieved 2017.
  3. ^ Glaisher, James Whitbread Lee (September 1871). "On a class of definite integrals. Part II". London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 4. 42 (279): 421-436. doi:10.1080/14786447108640600. Retrieved 2017.
  4. ^ "A007680 - OEIS". oeis.org. Retrieved 2020.
  5. ^ Weisstein, Eric W. "Erf". MathWorld. Wolfram.
  6. ^ H. M. Schöpf and P. H. Supancic, "On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion," The Mathematica Journal, 2014. doi:10.3888/tmj.16-11.Schöpf, Supancic
  7. ^ Weisstein, E. W. "Bürmann's Theorem". Wolfram MathWorld--A Wolfram Web Resource.
  8. ^ Bergsma, Wicher (2006). "On a new correlation coefficient, its orthogonal decomposition and associated tests of independence". arXiv:math/0604627.
  9. ^ Cuyt, Annie A. M.; Petersen, Vigdis B.; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). Handbook of Continued Fractions for Special Functions. Springer-Verlag. ISBN 978-1-4020-6948-2.
  10. ^ Schlömilch, Oskar Xavier (1859). "Ueber facultätenreihen". Zeitschrift für Mathematik und Physik (in German). 4: 390-415. Retrieved 2017.
  11. ^ Eq (3) on page 283 of Nielson, Niels (1906). Handbuch der Theorie der Gammafunktion (in German). Leipzig: B. G. Teubner. Retrieved 2017.
  12. ^ Chiani, M.; Dardari, D.; Simon, M.K. (2003). "New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels" (PDF). IEEE Transactions on Wireless Communications. 2 (4): 840-845. CiteSeerX 10.1.1.190.6761. doi:10.1109/TWC.2003.814350.
  13. ^ Tanash, I.M.; Riihonen, T. (2020). "Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials". IEEE Transactions on Communications. 68 (10): 6514-6524. arXiv:2007.06939. doi:10.1109/TCOMM.2020.3006902. S2CID 220514754.
  14. ^ Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]". Zenodo. doi:10.5281/zenodo.4112978.
  15. ^ Karagiannidis, G. K., & Lioumpas, A. S. An improved approximation for the Gaussian Q-function. 2007. IEEE Communications Letters, 11(8), pp. 644-646.
  16. ^ Chang, Seok-Ho; Cosman, Pamela C.; Milstein, Laurence B. (November 2011). "Chernoff-Type Bounds for the Gaussian Error Function". IEEE Transactions on Communications. 59 (11): 2939-2944. doi:10.1109/TCOMM.2011.072011.100049. S2CID 13636638.
  17. ^ Winitzki, Serge (2003). "Uniform approximations for transcendental functions". Lecture Notes in Comput. Sci. Lecture Notes in Computer Science. 2667. Spronger, Berlin. pp. 780-789. doi:10.1007/3-540-44839-X_82. ISBN 978-3-540-40155-1. (Sect. 3.1 "Error Function of Real Argument erf x")
  18. ^ Zeng, Caibin; Chen, Yang Cuan (2015). "Global Padé approximations of the generalized Mittag-Leffler function and its inverse". Fractional Calculus and Applied Analysis. 18 (6): 1492-1506. arXiv:1310.5592. doi:10.1515/fca-2015-0086. S2CID 118148950. Indeed, Winitzki [32] provided the so-called global Padé approximation
  19. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse". Cite journal requires |journal= (help)
  20. ^ Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 0-521-43064-X), 1992, page 214, Cambridge University Press.
  21. ^ a b c Cody, W. J. (March 1993), "Algorithm 715: SPECFUN--A portable FORTRAN package of special function routines and test drivers" (PDF), ACM Trans. Math. Softw., 19 (1): 22-32, CiteSeerX 10.1.1.643.4394, doi:10.1145/151271.151273, S2CID 5621105
  22. ^ Zaghloul, M. R. (1 March 2007), "On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand", Monthly Notices of the Royal Astronomical Society, 375 (3): 1043-1048, doi:10.1111/j.1365-2966.2006.11377.x
  23. ^ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations Archived 3 April 2012 at the Wayback Machine, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571-575.
  24. ^ Behnad, Aydin (2020). "A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis". IEEE Transactions on Communications. 68 (7): 4117-4125. doi:10.1109/TCOMM.2020.2986209. S2CID 216500014.
  25. ^ Carslaw, H. S.; Jaeger, J. C. (1959), Conduction of Heat in Solids (2nd ed.), Oxford University Press, ISBN 978-0-19-853368-9, p 484
  26. ^ https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/math.h.html
  27. ^ https://www.gnu.org/software/gsl/doc/html/specfunc.html#error-functions

Further reading

External links


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