Error Function Complex
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Faddeeva Function
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Imaginary Error Function
Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by (1)
Complex Error Function C++
Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function error function complex argument of the first kind. For , (5) where is the incomplete gamma function. Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). For , (11) (12) Using integration by parts gives (13) (14) (15) (16) so (17) and continuing the procedure gives the asymptotic series (18) (19) (20) (OEIS A001147 and A000079). Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as (25) (26) Its derivative is (27) where is a Hermite polynomial. The first derivative is (28) and the integral is (29) Min Max Re Im Erf can also be extended to the complex plane, as illustrated above. A simple integral involving erf that Wolfram Language cannot do is given by (30) (M.R.D'Orsogna, pers. comm., May 9, 2004). More complicated integr
toolboxes, and other File Exchange content using Add-On Explorer in MATLAB. » Watch video Highlights from Error function of complex numbers erfz.m View all files Join the 15-year community celebration. Play games and win prizes! » Learn more complex gamma function 4.6 4.6 | 5 ratings Rate this file 14 Downloads (last 30 days) File error function values Size: 59.4 KB File ID: #18312 Version: 1.0 Error function of complex numbers by Marcel Leutenegger Marcel Leutenegger (view profile) 13 complex normal distribution files 65 downloads 4.2475 14 Jan 2008 (Updated 14 Jan 2008) Extend the error function to the complex plane. | Watch this File File Information Description This package contains two MATLAB functions e=ERF(r) and http://mathworld.wolfram.com/Erf.html e=ERFZ(z)} as MEX-files for Windows. ERF overloads the default MATLAB error function of real-valued numbers with a much faster implementation. ERFZ enhances ERF to evaluate the error function of complex numbers too. If called with real numbers, it is identical to ERF and equally fast. ERFZ can replace ERF if no error message is required when called with complex numbers. For compatibility with operating systems other than Windows on x86 processors, https://www.mathworks.com/matlabcentral/fileexchange/18312-error-function-of-complex-numbers ERFZ is egally implemented as a normal M-file, which relies upon the default ERF by MATLAB. Implementation details are found in the attached PDF manual. MATLAB release MATLAB 6.1 (R12.1) Other requirements x86 Windows platform (MEX-files); none (M-file). Tags for This File Please login to tag files. erferror functionmathematicsnumerical evaluation Cancel Please login to add a comment or rating. Comments and Ratings (7) 21 Aug 2015 Karan Gill Karan Gill (view profile) 0 files 0 downloads 0.0 An alternative is to use the Symbolic Math Toolbox if you have it. Ex: >> double(erf(sym(1+1i))) ans = 1.3162 + 0.1905i You could define an anonymous function to make it easier: >> erfCmplx = @(x) double(erf(sym(x))) erfCmplx = @(x)double(erf(sym(x))) >> erfCmplx(1+1i) ans = 1.3162 + 0.1905i Comment only 05 Nov 2012 Steven G. Johnson Steven G. Johnson (view profile) 1 file 25 downloads 4.75 Although this implementation is very good (and is competitive with Per's code in performance on my machine), note that it computes real(erf(z)) inaccurately near the imaginary z axis. e.g. real(erfz(1e-8 + 1i)) gives 3.1023...e-8, but the correct answer is 3.067...e-8 according to Mathematica. See http://ab-initio.mit.edu/Faddeeva for an alternative (free/open-source) function that is a compiled MEX plugin (hence running several times faster than this code) which achieves arou
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