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# Imaginary Error Function Wiki

## Contents

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. However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1 ⁡ ( z ) = ∑ k = 0 W. You can then call various functions. check over here

Anal. 7 (1), pp. 187–198 (1970). References Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.9. For , (5) where is the incomplete gamma function. Use sym to convert complex infinities to symbolic objects:[erfi(sym(i*inf)), erfi(sym(-i*inf))]ans = [ 1i, -1i]Handling Expressions That Contain Imaginary Error Function Many functions, such as diff and int, can handle expressions containing

## Complementary Error Function

Zaghloul and Ahmed N. MathCAD provides both erf(x) and erfc(x) for real arguments. For |x| near zero, F(x) ≈ x, and for |x| large, F(x) ≈ 1/(2x).

• 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.
• Negative integer values of Im(ƒ) are shown with thick red lines.
• Kramp (1799) used this integral for the definition of the complementary error function .
• It does not have branch cuts or branch points.
• F(x) satisfies the differential equation d F d x + 2 x F = 1 {\displaystyle {\frac {dF}{dx}}+2xF=1\,\!} with the initial conditionF(0)=0.
• Washington, DC: Hemisphere, pp.385-393, 1987.
• IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF
• Level of Im(ƒ) = 0 is shown with a thick green line.

Cambridge, England: Cambridge University Press, 1990. 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 From Wikipedia, the free encyclopedia Jump to: navigation, search Plot of the error function In mathematics, the error function (also called the Gauss error function) is a special function Error Function Matlab Assuming you have a C++ compiler and the mkoctfile command installed (mkoctfile comes with Octave, possibly in an octave-devel or similarly named package in GNU/Linux distributions), you can simply run make

Wolfram|Alpha» Explore anything with the first computational knowledge engine. Error Function Calculator Maple: Maple implements both erf and erfc for real and complex arguments. Practice online or make a printable study sheet. http://mathworld.wolfram.com/Erfi.html Intermediate levels of Im(ƒ) = constant are shown with thin green lines.

Prudnikov, A.P.; Brychkov, Yu.A.; and Marichev, O.I. Error Function Excel C++: C++11 provides erf() and erfc() in the header cmath. V . ∫ − ∞ ∞ e − a x 2 y − x d x {\displaystyle H_{a}=\pi ^{-1}P.V.\int _{-\infty }^{\infty }{e^{-ax^{2}} \over y-x}dx} The nth derivative is ∂ n H which follows easily by induction, writing and integrating by parts.

## Error Function Calculator

MathCAD provides both erf(x) and erfc(x) for real arguments. The Q-function can be expressed in terms of the error function as The inverse of is known as the normal quantile function, or probit function and may be expressed in terms Complementary Error Function 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 ) = Error Function Table The inverse of the generalized error function is an analytical function of and , which is defined in .

Translate erfiImaginary error functioncollapse all in page Syntaxerfi(x) exampleDescriptionexampleerfi(x) returns the imaginary error function of x. check my blog 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 Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED functions to calculate the error function and its complement for real arguments. Wolfram Language» Knowledge-based programming for everyone. Inverse Error Function

R. See Terms of Use for details. 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 this content Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. 8 of 8 04-03-2011 11:22 AM RECOMMENDED Spirometry - Wikipedia, The Free Encyclopedia Tris - Wikipedia, The Free

For convenience, a script to compile all of the plugins using the mex command in Matlab is included. Error Function Python Sloane, N.J.A. 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.

## Springer-Verlag.

Analytic Theory of Continued Fractions. 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 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 ) Error Function Properties If L is sufficiently far from the mean, i.e. μ − L ≥ σ ln ⁡ k {\displaystyle \mu -L\geq \sigma {\sqrt {\ln {k}}}} , then: Pr [ X ≤ L

Using the alternate value a≈0.147 reduces the maximum error to about 0.00012.This approximation can also be inverted to calculate the inverse error function: A pure exponential approximation for the complementary error MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. In benchmarks of our code, we find that it is comparable to or faster than most competing software for these functions in the complex plane (but we also have special-case optimizations http://renderq.net/error-function/imaginary-error-function-values.php Soft. 16 (1), pp. 38–46 (1990); this is TOMS Algorithm 680.

Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Matlab provides both erf and erfc for real arguments, also via W. Thus π 1 / 2 H ( y ) = Im ⁡ ∫ 0 ∞ d k exp ⁡ [ − k 2 / 4 + i k y ] {\displaystyle

We see this with the technique of differentiating inside the integral sign. 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", 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 For , (11) (12) Using integration by parts gives (13) (14) (15) (16) so (17) and continuing the procedure gives the asymptotic series (18) (19) (20) (OEIS A001147 and A000079).

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 This usage is similar to the Q-function, which in fact can be written in terms of the error function.PropertiesThe property means that the error function is an odd function.For any complex At the imaginary axis, it tends to ±i∞.Taylor seriesThe error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.The defining integral