Error Function Complex Number
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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
Hyperbolic Function Of Complex Number
∫ 0 x e − t 2 d t . {\displaystyle {\begin − 6\operatorname exponential function in complex number − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _
Analytic Function In Complex Numbers
9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π excel complex number functions ∫ x ∞ e − t 2 d t = e − 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]). complementary error function Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} 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 us
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Error Function Calculator
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Error Function Table
in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top erf(a+ib) https://en.wikipedia.org/wiki/Error_function error function separate into real and imaginary part up vote 5 down vote favorite 3 Is there an easy way to separate erf(a+ib) into real and imaginary part? calculus integration complex-analysis contour-integration share|cite|improve this question edited Mar 14 '14 at 22:49 Ron Gordon 109k12130221 asked Mar 14 '14 at 19:04 Sleepyhead 1385 add a comment| 3 Answers 3 active oldest votes up vote 6 down vote I'm not sure if you are interested in http://math.stackexchange.com/questions/712434/erfaib-error-function-separate-into-real-and-imaginary-part an analytical answer or a computational answer; these are two different things. The analytical answer is...not really, unless you consider GEdgar's answer useful. (And one might.) The computational answer is a resounding yes. A result found in Abramowitz & Stegun claims the following: $$\operatorname*{erf}(x+i y) = \operatorname*{erf}{x} + \frac{e^{-x^2}}{2 \pi x} [(1-\cos{2 x y})+i \sin{2 x y}]\\ + \frac{2}{\pi} e^{-x^2} \sum_{k=1}^{\infty} \frac{e^{-k^2/4}}{k^2+4 x^2}[f_k(x,y)+i g_k(x,y)] + \epsilon(x,y) $$ where $$f_k(x,y) = 2 x (1-\cos{2 x y} \cosh{ k y}) + k\sin{2 x y} \sinh{k y}$$ $$g_k(x,y) = 2 x \sin{2 x y} \cosh{k y} + k\cos{2 x y} \sinh{k y}$$ Then $$\left |\epsilon(x,y) \right | \le 10^{-16} |\operatorname*{erf}{(x+i y)}| $$ This accuracy is valid for all $x$ and $y$, i.e., the complex plane. I will present a derivation of this result to show you where the error term comes from. Consider the definition of the error function in the complex plane: $$\operatorname*{erf}{z} = \frac{2}{\sqrt{\pi}} \int_{\Gamma} d\zeta \, e^{-\zeta^2}$$ where $\Gamma$ is any path in the complex plane from $\zeta = 0$ to $\zeta=z$. Consider, then, the special case where $\Gamma$ is the path that runs from $0$ to $x$ along the real axis, then from $x$ to $z=x+i y$ parallel to the imaginary axis. Seen this way, the error function of a complex number is equal to $$\operatorname*{erf}{(x+i y)} = \
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 4.6 4.6 | https://www.mathworks.com/matlabcentral/fileexchange/18312-error-function-of-complex-numbers 5 ratings Rate this file 14 Downloads (last 30 days) File Size: 59.4 KB File ID: #18312 Version: 1.0 Error function of complex numbers by Marcel Leutenegger Marcel Leutenegger (view profile) 13 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 e=ERFZ(z)} as MEX-files for Windows. error function 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, ERFZ is egally implemented as a normal function in complex 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 around 13 digits of accuracy or more in both the real and imaginary parts