Fortran Imaginary Error Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t error function integral 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int error function calculator _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end error function table 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e −
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x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} inverse error function is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} 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: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{
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! complementary error function table » Learn more 4.6 4.6 | 5 ratings Rate this file 11 Downloads
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(last 30 days) File Size: 59.4 KB File ID: #18312 Version: 1.0 Error function of complex numbers by Marcel Leutenegger
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Marcel Leutenegger (view profile) 13 files 62 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 https://en.wikipedia.org/wiki/Error_function package contains two MATLAB functions e=ERF(r) and 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 https://www.mathworks.com/matlabcentral/fileexchange/18312-error-function-of-complex-numbers numbers. For compatibility with operating systems other than Windows on x86 processors, 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 24 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 Mathemat
type shall error function be REAL. Return value:The return value is of type REAL, of the same kind error function table as X and lies in the range -1 \leq erf (x) \leq 1 . Example: program test_erf real(8) :: x = 0.17_8 x = erf(x) end program test_erf Specific names: Name Argument Return type Standard DERF(X) REAL(8) X REAL(8) GNU extension