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# Error Function Complex Argument

## Contents

Instead of using definite integrals, the three univariate error functions can be defined through the following infinite series.

A quick look at the probability integrals and inversesHere is a quick New York: Random House, 1963. and Stegun, I.A. (Eds.). "Error Function and Fresnel Integrals." Ch.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1 ⁡ ( z ) = ∑ k = 0 useful reference

and Robinson, G. "The Error Function." §92 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. I think these may be zeroes of the error function, but I have not yet investigated.) And now, here's a plot of some results; note that the plot of the effective Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. ERFZ enhances ERF to evaluate the error function of complex numbers too.

## Error Function Complex Argument

Retrieved 2011-10-03. ^ Chiani, M., Dardari, D., Simon, M.K. (2003). Similarly, (8) (OEIS A103979 and A103980). Wolfram Language» Knowledge-based programming for everyone.

1. Softw., 19 (1): 22–32, doi:10.1145/151271.151273 ^ Zaghloul, M.
2. These generalised functions can equivalently be expressed for x>0 using the Gamma function and incomplete Gamma function: E n ( x ) = 1 π Γ ( n ) ( Γ

Fortran 77 implementations are available in SLATEC. That's why this result is a big deal: analytically, it is not equal to the error function, but computationally, it is equal for all practical purposes. 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) Error Function Values In particular cases, these formulas can be simplified to the following relations: Integral representations The probability integrals , , , and can also be represented through the following equivalent integrals: The

The inverse complementary error function is defined as erfc − 1 ⁡ ( 1 − z ) = erf − 1 ⁡ ( z ) . {\displaystyle \operatorname ζ 8 ^{-1}(1-z)=\operatorname Gamma Function Complex Argument D: A D package[16] exists providing efficient and accurate implementations of complex error functions, along with Dawson, Faddeeva, and Voigt functions. The system returned: (22) Invalid argument The remote host or network may be down. http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html Thus, I have implemented a simple while loop to achieve a desired precision.

Kramp (1799) used this integral for the definition of the complementary error function . Complementary Error Function share|cite|improve this answer edited Oct 1 '15 at 13:33 answered Mar 14 '14 at 21:24 Ron Gordon 109k12130221 There is no $a$ on the LHS of your last approximation. 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 and Oldham, K.B. "The Error Function and Its Complement ." Ch.40 in An Atlas of Functions.

## Gamma Function Complex Argument

Computerbasedmath.org» Join the initiative for modernizing math education. New York: Chelsea, 1948. Error Function Complex Argument 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)] Delta Function Complex Argument To obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed.

ContourPlot[-Log[10, Abs[g[x, y, 0.5]/(Erf[x + I y] + 10^(-16)) - 1]], {x, -2, 2}, {y, -4, 4}, PlotPoints -> 20, PlotLegends -> Automatic] The high amount of detail is indicative of see here How can there be different religions in a world where gods have been proven to exist? See [2]. ^ http://hackage.haskell.org/package/erf ^ Commons Math: The Apache Commons Mathematics Library ^ a b c Cody, William J. (1969). "Rational Chebyshev Approximations for the Error Function" (PDF). Derivative and integral The derivative of the error function follows immediately from its definition: d d z erf ⁡ ( z ) = 2 π e − z 2 . {\displaystyle Complex Error Function Matlab

Also has erfi for calculating i erf ⁡ ( i x ) {\displaystyle i\operatorname {erf} (ix)} Maple: Maple implements both erf and erfc for real and complex arguments. Numerical approximations Over the complete range of values, there is an approximation with a maximal error of 1.2 × 10 − 7 {\displaystyle 1.2\times 10^{-7}} , as follows:[15] erf ⁡ ( Parity and symmetry The probability integrals , , and are odd functions and have mirror symmetry: The generalized error function has permutation symmetry: The complementary error function has mirror symmetry: Series this page Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian

## The system returned: (22) Invalid argument The remote host or network may be down.

For any complex number z: erf ⁡ ( z ¯ ) = erf ⁡ ( z ) ¯ {\displaystyle \operatorname − 0 ({\overline ⁡ 9})={\overline {\operatorname ⁡ 8 (z)}}} where z C# Optional Array Parameter for Class A power source that would last a REALLY long time Computational chemistry: research in organic chemistry? Java: Apache commons-math[19] provides implementations of erf and erfc for real arguments. Inverse Error Function Note that you can use other values of $a$ than $1/2$ so you can tune the number of terms you need from the infinite sum.

MR0167642. Go: Provides math.Erf() and math.Erfc() for float64 arguments. 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 Get More Info Not the answer you're looking for?

Why are so many metros underground? e.g. A simple integral involving erf that Wolfram Language cannot do is given by (30) (M.R.D'Orsogna, pers. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation.

Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). How to remove a stuck (maybe melted) connector from the blower motor resistor Antonym for the word "hero" not in the sense of "villain" What if my company didn't pay the Perl: erf (for real arguments, using Cody's algorithm[20]) is implemented in the Perl module Math::SpecFun Python: Included since version 2.7 as math.erf() and math.erfc() for real arguments. Haskell: An erf package[18] exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

There is also some structure around where the computation was not able to achieve the desired level of precision; again, this is worth investigating. All generalised error functions for n>0 look similar on the positive x side of the graph. ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF). Princeton, NJ: Princeton University Press, p.105, 2003.

A two-argument form giving is also implemented as Erf[z0, z1]. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

Contact us MathWorks Accelerating the pace of engineering and science MathWorks is the leading developer of mathematical computing software for engineers and scientists. MATLAB release MATLAB 6.1 (R12.1) Other requirements x86 Windows platform (MEX-files); none (M-file). Integrals and Series, Vol.2: Special Functions. Transformations If the arguments of the probability integrals , , and contain square roots, the arguments can sometimes be simplified: Representations of derivatives The derivative of the probability integrals , ,

Supancic, "On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion," The Mathematica Journal, 2014.