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# Imaginary Error Function Wiki

## Contents

The complex functions are Faddeeva_erf(double complex z, double relerr) etc. More specifically, near the origin it has the series expansion F ( x ) = ∑ k = 0 ∞ ( − 1 ) k 2 k ( 2 k + Wolfram Language» Knowledge-based programming for everyone. Sequences A000079/M1129, A001147/M3002, and A084253 in "The On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|Alpha: Erfi CITE THIS AS: Weisstein, Eric W. "Erfi." From MathWorld--A Wolfram Web Resource. Check This Out

J. Compute the imaginary error function for these numbers. This is useful, for example, in determining the bit error rate of a digital communication system. Compute the imaginary error function for elements of matrix M and vector V:M = sym([0 inf; 1/3 -inf]); V = sym([1; -i*inf]); erfi(M) erfi(V)ans = [ 0, Inf] [ erfi(1/3), -Inf]

## Complementary Error Function

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 ⁡ ( Google search: Google's search also acts as a calculator and will evaluate "erf(...)" and "erfc(...)" for real arguments. V . ∫ − ∞ ∞ x 2 n e − x 2 y − x d x {\displaystyle H_{n}=\pi ^{-1}P.V.\int _{-\infty }^{\infty }{x^{2n}e^{-x^{2}} \over y-x}dx} Introduce H a = π Math.

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. Translate erfiImaginary error functioncollapse all in page Syntaxerfi(x) exampleDescriptionexampleerfi(x) returns the imaginary error function of x. 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 Error Function Matlab For iterative calculation of the above series, the following alternative formulation may be useful: because term).

Positive integer values of Im(f) are shown with thick blue lines. Error Function Calculator Cody's rational Chebyshev approximation algorithm.[20] Ruby: Provides Math.erf() and Math.erfc() for real arguments. A change of variable also gives H a = 2 π − 1 / 2 F ( y a ) {\displaystyle H_{a}=2\pi ^{-1/2}F(y{\sqrt {a}})} . http://mathworld.wolfram.com/Erfi.html Contact the MathWorld Team © 1999-2016 Wolfram Research, Inc. | Terms of Use THINGS TO TRY: erf error function (1) error function (0) Erfi Gamma, Beta, Erf Erfi[z] Introduction

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. Error Function Excel A Makefile is included. After division by n!, all the En for odd n look similar (but not identical) to each other. The Hilbert transform of x 2 n e − x 2 {\displaystyle x^{2n}e^{-x^{2}}} is also related to the Dawson function.

1. 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",
2. 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 =
3. In particular, we did not refer to the authors' Fortran or Matlab implementations (respectively), which are under restrictive "semifree" ACM copyright terms and are therefore unusable in free/open-source software.
4. However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1 ⁡ ( z ) = ∑ k = 0
5. Soft. 38 (2), 15 (2011).
6. 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
7. Indeed, or rearranged for erf and erfc: Consequently, the error function is also closely related to the Q-function, which is the tail probability of the standard normal distribution.
8. To obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed.
9. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions.
10. Ruby: Provides Math.erf() and Math.erfc() for real arguments.

## Error Function Calculator

Applications When the results of a series of measurements are described by a normal distribution with standard deviation σ {\displaystyle \textstyle \sigma } and expected value 0, then erf ( a http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Complementary Error Function Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha. Error Function Table It is an essential singular point.

Math. his comment is here Excel: Microsoft Excel provides the erf, and the erfc functions, nonetheless both inverse functions are not in the current library. Indeed, the exact value of the remainder is which follows easily by induction, writing and integrating by parts.For large enough values of x, only the first few terms of this asymptotic J. (March 1993), "Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers" (PDF), ACM Trans. Inverse Error Function

Using the alternate value a≈0.147 reduces the maximum error to about 0.00012.[12] This approximation can also be inverted to calculate the inverse error function: erf − 1 ⁡ ( x ) References Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.9. Wolfram|Alpha» Explore anything with the first computational knowledge engine. this contact form Excel: Microsoft Excel provides the erf, and the erfc functions, nonetheless both inverse functions are not in the current library.[17] Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED

For previous versions or for complex arguments, SciPy includes implementations of erf, erfc, erfi, and related functions for complex arguments in scipy.special. Error Function Python For , (5) where is the incomplete gamma function. For convenience, a script to compile all of the plugins using the mex command in Matlab is included.

## ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF).

has derivative (2) and integral (3) It has series about given by (4) (where the terms are OEIS A084253), and series about infinity given by (5) (OEIS A001147 and A000079). In some cases, however, there are additional complications that require our implementation to go beyond these simple formulas. The imaginary error function has a very similar Maclaurin series, which is: erfi ⁡ ( z ) = 2 π ∑ n = 0 ∞ z 2 n + 1 n Error Function Properties in the vicinity of |z|=1 [although comparison with other compilers suggests that this may be a problem specific to gfortran].

The integrand ƒ = exp(−z2) and ƒ = erf(z) are shown in the complex z-plane in figures 2 and 3. This series diverges for every finite x, and its meaning as asymptotic expansion is that, for any N ∈ N {\displaystyle N\in \mathbb Γ 2 } one has erfc ⁡ ( For erf, large cancellation errors occur in these formulas near |z|=0 where w(z) is nearly 1, as well as near the imaginary axis for Re[erf], and in these regimes we switch navigate here H.

In order of increasing accuracy, they are: (maximum error: 5×10−4) where a1=0.278393, a2=0.230389, a3=0.000972, a4=0.078108 (maximum error: 2.5×10−5) where p=0.47047, a1=0.3480242, a2=−0.0958798, a3=0.7478556 (maximum error: 3×10−7) where a1=0.0705230784, a2=0.0422820123, a3=0.0092705272, a4=0.0001520143, Softw., 19 (1): 22–32, doi:10.1145/151271.151273 ^ Zaghloul, M. Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 6.2. 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

and on MathWorld (http://mathworld.wolfram.com/ProbabilityIntegral.html) 2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (http://homepages.physik.uni-muenchen.de/~Winitzki/erf-approx.pdf) (PDF). Weisstein ^ Bergsma, Wicher. "On a new correlation coefficient, its orthogonal decomposition and associated tests of independence" (PDF). ^ Cuyt, Annie A. Math. Similarly, D − ( x ) = π 2 e x 2 e r f ( x ) {\displaystyle D_{-}(x)={\frac {\sqrt {\pi }}{2}}e^{x^{2}}\mathrm {erf} (x)} in terms of the real error

In order of increasing accuracy, they are: erf ⁡ ( x ) ≈ 1 − 1 ( 1 + a 1 x + a 2 x 2 + a 3 x Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha. Conf., vol. 2, pp. 571–575. ^ Van Zeghbroeck, Bart; Principles of Semiconductor Devices, University of Colorado, 2011. [1] ^ Wolfram MathWorld ^ H. W.

Integrand exp(−z2) Taylor series The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.