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# Imaginary Error Function Wikipedia

## Contents

Online Integral Calculator» Solve integrals with Wolfram|Alpha. Hints help you try the next step on your own. MathCAD provides both erf(x) and erfc(x) for real arguments. The imaginary error function has special values for these parameters: [erfi(sym(0)), erfi(sym(inf)), erfi(sym(-inf))]ans = [ 0, Inf, -Inf]Compute the imaginary error function for complex infinities. check over here

Because these numbers are not symbolic objects, you get floating-point results.s = [erfi(1/2), erfi(1.41), erfi(sqrt(2))]s = 0.6150 3.7382 3.7731Compute the imaginary error function for the same numbers converted to symbolic objects. IDL: provides both erf and erfc for real and complex arguments. For most symbolic (exact) numbers, erfi returns unresolved symbolic calls.s = [erfi(sym(1/2)), erfi(sym(1.41)), erfi(sqrt(sym(2)))]s = [ erfi(1/2), erfi(141/100), erfi(2^(1/2))]Use vpa to approximate this result with the 10-digit accuracy:vpa(s, 10)ans = [ 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

## Complementary Error Function

1. In terms of either erfi or the Faddeeva function w(z), the Dawson function can be extended to the entire complex plane:[2] F ( z ) = π 2 e − z
2. R: Baptiste Auguié and Dirk Eddelbuettel have written an R interface to the Faddeeva package, available on Github and on CRAN.
3. Similarly, (8) (OEIS A103979 and A103980).
4. For |x| near zero, F(x) ≈ x, and for |x| large, F(x) ≈ 1/(2x).

See alsoRelated functionsGaussian integral, over the whole real line Gaussian function, derivative Dawson function, renormalized imaginary error function In probabilityNormal distribution Normal cumulative distribution function, a scaled and shifted form of The inverse of the generalized error function is an analytical function of and , which is defined in . Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Error Function Matlab THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.

Erf has the continued fraction (32) (33) (Wall 1948, p.357), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and rediscovered by Ramanujan (Watson 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 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 = my review here The Hilbert transform of x 2 n e − x 2 {\displaystyle x^{2n}e^{-x^{2}}} is also related to the Dawson function.

Install the resulting *.mex* files, along with the *.m help files, into your Matlab path GNU Octave: Similar to Matlab, above, we provide source code for compiled GNU Octave plugins (.oct Error Function Excel Analytic Theory of Continued Fractions. Arfken, G. 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 ⁡ (

## Error Function Calculator

Laplace (1812) derived an asymptotic expansion of the error function. http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html External linksMathWorld – Erf Error-function numerical table and calculator

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Toggle Main Navigation Log In Products Solutions Academia Support Community Events Complementary Error Function By using this site, you agree to the Terms of Use and Privacy Policy. Error Function Table Simple values at zero and one The probability integrals , , , and , and their inverses , , and have simple values for zero or unit arguments: Simple values at

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 http://renderq.net/error-function/imaginary-error-function-values.php For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). Whittaker, E.T. The inverse was systematically investigated in the second half of the twentieth century, especially by J. Inverse Error Function

Wolfram Language» Knowledge-based programming for everyone. This usage is similar to the Q-function, which in fact can be written in terms of the error function. Continued fraction expansion A continued fraction expansion of the complementary error function is:[11] erfc ⁡ ( z ) = z π e − z 2 1 z 2 + a 1 this content Interestingly, closed-form expressions for the truncated version of the Taylor series at the origin can be expressed through generalized hypergeometric function , for example: Asymptotic series expansions The asymptotic behavior of

References Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.9. Error Function Python Fortran 77 implementations are available in SLATEC. Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha.

## G.

It is implemented in the Wolfram Language as Erfi[z]. Johnson has written free/open-source C++ code (with wrappers for C, Matlab, GNU Octave, Python, R, Scilab, and Julia) to compute the various error functions of arbitrary complex arguments. A complex-argument erf is also in the arbitrary-precision arithmetic mpmath library as mpmath.erf() R: "The so-called 'error function'" is not provided directly, but is detailed as an example of the normal Error Function Properties R.

Algorithm 916 requires an external complementary error function erfc(x) function for real arguments x to be supplied as a subroutine. Cody's rational Chebyshev approximation algorithm.[20] Ruby: Provides Math.erf() and Math.erfc() for real arguments. Matlab provides both erf and erfc for real arguments, also via W. have a peek at these guys MathCAD provides both erf(x) and erfc(x) for real arguments.

Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. Related functions The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by software languages, as they differ only by scaling and translation. 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] Haskell: An erf package[18] exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

Hardy, G.H. MR0167642. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. 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