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# Imaginary Error Function Wiki

## Contents

For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). Simple values at zero and one The probability integrals , , , and , and their inverses , , and have simple values for zero or unit arguments: Simple values at ISBN0-486-61272-4. 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 http://colvertgroup.com/error-function/imaginary-error-function.php

Kramp (1799) used this integral for the definition of the complementary error function . 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: w ( z ) = instead of Faddeeva::erf, and the real-argument versions are Faddeeva_erf_re(double x) etc. (Note that in gcc you may need to compile with the -std=c99 flag to enable C99 support.) Matlab (also available Princeton, NJ: Princeton University Press, p.105, 2003. https://en.wikipedia.org/wiki/Error_function

## Complementary Error Function

To compute the Faddeeva function for sufficiently large |z|, we use a continued-fraction expansion for w(z) similar to those described in Walter Gautschi, "Efficient computation of the complex error function," SIAM In order of increasing accuracy, they are: erf ⁡ ( x ) ≈ 1 − 1 ( 1 + a 1 x + a 2 x 2 + a 3 x However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1 ⁡ ( z ) = ∑ k = 0

1. Another approximation is given by erf ⁡ ( x ) ≈ sgn ⁡ ( x ) 1 − exp ⁡ ( − x 2 4 π + a x 2 1
2. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans.
3. Integrand exp(−z2) Taylor series The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.
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5. ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF).
6. Acton, F.S.
7. The error function is related to the cumulative distribution Φ {\displaystyle \Phi } , the integral of the standard normal distribution, by[2] Φ ( x ) = 1 2 + 1
8. Preprint available at arXiv:1106.0151. (I initially used this algorithm for all z, but the continued-fraction expansion turned out to be faster for larger |z|.
10. H.

H ( y ) {\displaystyle H(y)} can be related to the Dawson function as follows. Ruby: Provides Math.erf() and Math.erfc() for real arguments. Hints help you try the next step on your own. Error Function Matlab 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

P.‐S. Error Function Excel Conf., vol. 2, pp. 571–575. ^ Van Zeghbroeck, Bart; Principles of Semiconductor Devices, University of Colorado, 2011. [1] ^ Wolfram MathWorld ^ H. Go: Provides math.Erf() and math.Erfc() for float64 arguments. Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680).

## Error Function Calculator

The inverse of the generalized error function is an analytical function of and , which is defined in . http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html A complex generalization of is defined as (39) (40) Integral representations valid only in the upper half-plane are given by (41) (42) SEE ALSO: Dawson's Integral, Erfc, Erfi, Fresnel Integrals, Gaussian Complementary Error Function See Terms of Use for details. Error Function Table http://homepages.physik.uni-muenchen.de /~Winitzki/erf-approx.pdf. 3. ^ http://docs.scipy.org/doc/scipy/reference/generated/scipy.special.erf.html 4. ^ http://hackage.haskell.org/package/erf Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 7" (http://www.math.sfu.ca/~cbm/aands /page_297.htm) , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New

Are you sure you want to continue?CANCELOKWe've moved you to where you read on your other device.Get the full title to continueGet the full title to continue reading from where you http://colvertgroup.com/error-function/imaginary-error-function-matlab.php Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. References Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.9. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 Chiani, M.; Dardari, D.; Simon, M. Inverse Error Function

Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha. Positive integer values of Im(f) are shown with thick blue lines. The complex functions are Faddeeva_erf(double complex z, double relerr) etc. navigate here doi:10.1090/S0025-5718-1969-0247736-4. ^ Error Function and Fresnel Integrals, SciPy v0.13.0 Reference Guide. ^ R Development Core Team (25 February 2011), R: The Normal Distribution Further reading Abramowitz, Milton; Stegun, Irene Ann, eds.

W. Error Function Python Comp. 23 (107): 631–637. More precisely, it requires the scaled function erfcx(x) = ex2erfc(x).