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# Imaginary Error Function Wikipedia

## Contents

ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF). The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. Fortran 77 implementations are available in SLATEC. This page has been accessed 57,852 times. http://colvertgroup.com/error-function/imaginary-error-function.php

External links gsl_sf_dawson in the GNU Scientific Library Cephes – C and C++ language special functions math library Faddeeva Package – C++ code for the Dawson function of both real and Contact the MathWorld Team © 1999-2016 Wolfram Research, Inc. | Terms of Use THINGS TO TRY: erf erfi x erfi (0) Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of You can also select a location from the following list: Americas Canada (English) United States (English) Europe Belgium (English) Denmark (English) Deutschland (Deutsch) España (Español) Finland (English) France (Français) Ireland (English) For any complex number z: erf ⁡ ( z ¯ ) = erf ⁡ ( z ) ¯ {\displaystyle \operatorname − 0 ({\overline ⁡ 9})={\overline {\operatorname ⁡ 8 (z)}}} where z

## Complementary Error Function

doi:10.1109/TCOMM.2011.072011.100049. ^ Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 0-521-43064-X), 1992, page 214, Cambridge University Press. ^ DlangScience/libcerf, A package for use with the D Programming language. Java: Apache commons-math[19] provides implementations of erf and erfc for real arguments. Cody's algorithm.

1. The inverse complementary error function is defined as erfc − 1 ⁡ ( 1 − z ) = erf − 1 ⁡ ( z ) . {\displaystyle \operatorname ζ 8 ^{-1}(1-z)=\operatorname
2. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
3. Math.
4. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.
5. This function is used in statistics to predict behavior of any sample with respect to the population mean.
6. Wolfram Language» Knowledge-based programming for everyone.
7. New York: Dover, pp.297-309, 1972.
8. Specifying a larger value of relerr may improve performance for some z (at the expense of accuracy).
9. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 ^ Dawson, H.
10. Online Integral Calculator» Solve integrals with Wolfram|Alpha.

Online Integral Calculator» Solve integrals with Wolfram|Alpha. On the other hand, Algorithm 916 is competitive or faster for smaller |z|, and appears to be significantly more accurate than the Poppe & Wijers code in some regions, e.g. Download the code and documentation from: http://ab-initio.mit.edu/Faddeeva-MATLAB.zip (a zip file) The provided functions are called Faddeeva_w, Faddeeva_erf, Faddeeva_erfc, Faddeeva_erfi, Faddeeva_erfcx, and Faddeeva_Dawson, equivalent to the C++ functions above. Error Function Matlab For real arguments , the values of the inverse error function are real; for real arguments , the values of the inverse of the generalized error function are real; and for

Cody's rational Chebyshev approximation algorithm.[20] Ruby: Provides Math.erf() and Math.erfc() for real arguments. Error Function Calculator Washington, DC: Hemisphere, pp.385-393, 1987. Haskell: An erf package exists that provides a typeclass for the error function and implementations for the native (real) floating point types. http://mathworld.wolfram.com/Erfi.html Assoc.

The inverse of the generalized error function is an analytical function of and , which is defined in . Error Function Excel erf, the error function erfc, the complementary error function erfcx, the scaled complementary error function erfi, the imaginary error function Dawson, the Dawson function Given the Faddeeva function w(z) and the and Robinson, G. "The Error Function." §92 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. This usage is similar to the Q-function, which in fact can be written in terms of the error function.

## Error Function Calculator

Practice online or make a printable study sheet. try this Wolfram|Alpha» Explore anything with the first computational knowledge engine. Complementary Error Function History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Error Function Table Both functions are overloaded to accept arguments of type float, double, and long double.

Omitting the relerr argument, or passing relerr=0 (or any relerr less than machine precision ε≈10−16), corresponds to requesting machine precision, and in practice a relative error <10−13 is usually achieved. http://colvertgroup.com/error-function/imaginary-error-function-matlab.php Compute the first and second derivatives of the imaginary error function:syms x diff(erfi(x), x) diff(erfi(x), x, 2)ans = (2*exp(x^2))/pi^(1/2) ans = (4*x*exp(x^2))/pi^(1/2)Compute the integrals of these expressions:int(erfi(x), x) int(erfi(log(x)), x)ans = The Hilbert transform of x 2 n e − x 2 {\displaystyle x^{2n}e^{-x^{2}}} is also related to the Dawson function. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Inverse Error Function

For example: extern std::complex Faddeeva::w(std::complex z, double relerr=0); This function Faddeeva::w(z, relerr) computes w(z) to a desired relative error relerr. http://mathworld.wolfram.com/Erf.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. G. (1897). "On the Numerical Value of ∫ 0 h exp ⁡ ( x 2 ) d x {\displaystyle \int _{0}^{h}\exp(x^{2})dx} ". his comment is here 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): erf ⁡ ( x ) = 2 x

Softw., 19 (1): 22–32, doi:10.1145/151271.151273 ^ Zaghloul, M. Error Function Python Both functions are overloaded to accept arguments of type float, double, and long double. The Dawson function is the one-sided Fourier-Laplace sine transform of the Gaussian function, D + ( x ) = 1 2 ∫ 0 ∞ e − t 2 / 4 sin

## denotes the Cauchy principal value, and we restrict ourselves to real y {\displaystyle y} .

After division by n!, all the En for odd n look similar (but not identical) to each other. For convenience, a script to compile all of the plugins using the mex command in Matlab is included. Similarly, (8) (OEIS A103979 and A103980). Error Function Properties The denominator terms are sequence A007680 in the OEIS.

Assuming you have a C++ compiler installed (and have run mex -setup to tell Matlab to use it), you can simply run the Faddeeva_build.m script in Matlab to compile all of Incomplete Gamma Function and Error Function", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8 Temme, Nico M. (2010), "Error Functions, Dawson’s and Fresnel Integrals", Wolfram|Alpha» Explore anything with the first computational knowledge engine. weblink Similarly, the erf, erfc, erfcx, erfi, and Dawson functions are computed by calling: extern std::complex Faddeeva::erf(std::complex z, double relerr=0); extern std::complex Faddeeva::erfc(std::complex z, double relerr=0); extern std::complex Faddeeva::erfcx(std::complex z, double relerr=0);

Fortran 77 implementations are available in SLATEC. LCCN64-60036. J. Strecok (1968).

Definitions of probability integrals and inverses The probability integral (error function) , the generalized error function , the complementary error function , the imaginary error function , the

SEE ALSO: Dawson's Integral, Erf, Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfi/ REFERENCES: Sloane, N.J.A. A Course in Modern Analysis, 4th ed.