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# Imaginary Error Function Wikipedia

## Contents

Inflection points follow for F ( x ) = x 2 x 2 − 1 {\displaystyle F(x)={\frac {x}{2x^{2}-1}}} , resulting in x=±1.50197526… ( A133843), F(x)=±0.42768661… ( A245262). (Apart from the trivial inflection point Math. The generalized error function is an analytical function of and , which is defined in . Weisstein. "Bürmann's Theorem" from Wolfram MathWorld—A Wolfram Web Resource./ E. Check This Out

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 ⁡ ( Alternatively, H n {\displaystyle H_{n}} can be calculated using the recurrence relation (for n ≥ 0 {\displaystyle n\geq 0} ) H n + 1 ( y ) = y 2 H New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels. The complex functions are Faddeeva_erf(double complex z, double relerr) etc.

## Complementary Error Function

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}})} . 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. http://mathworld.wolfram.com/Erfi.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. ISBN 978-0-486-61272-0.

Wijers, "More efficient computation of the complex error function," ACM Trans. H ( y ) {\displaystyle H(y)} can be related to the Dawson function as follows. The probability integrals were so named because they are widely applied in the theory of probability, in both normal and limit distributions. Error Function Matlab References Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.9.

W. Error Function Calculator Given random variable X ∼ Norm ⁡ [ μ , σ ] {\displaystyle X\sim \operatorname {Norm} [\mu ,\sigma ]} and constant L < μ {\displaystyle L<\mu } : Pr [ X 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. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

Taking the imaginary part of the result gives H ( y ) = 2 π − 1 / 2 F ( y ) {\displaystyle H(y)=2\pi ^{-1/2}F(y)} where F ( y ) Error Function Excel Because all of the algorithms are based on algorithms for the Faddeeva function, we call this the Faddeeva Package. Washington, DC: Math. A two-argument form giving is also implemented as Erf[z0, z1].

• Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha.
• 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
• However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1 ⁡ ( z ) = ∑ k = 0
• 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 = [
• doi:10.3888/tmj.16–11.Schöpf, Supancic ^ E.
• 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

## Error Function Calculator

Definite integrals involving include Definite integrals involving include (34) (35) (36) (37) (38) The first two of these appear in Prudnikov et al. (1990, p.123, eqns. 2.8.19.8 and 2.8.19.11), with , his comment is here Poles and essential singularities The probability integrals , , and have only one singular point at . Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 6.2. This page has been accessed 57,852 times. Inverse Error Function

Intermediate levels of Re(ƒ)=constant are shown with thin red lines for negative values and with thin blue lines for positive values. 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 The denominator terms are sequence A007680 in the OEIS. http://mttags.com/error-function/imaginary-error-function-infinity.php ISBN978-1-4020-6948-2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF).

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 Error Function Python Supancic, "On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion," The Mathematica Journal, 2014. Haskell: An erf package exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

## Sequences A000079/M1129, A001147/M3002, A007680/M2861, A103979, A103980 in "The On-Line Encyclopedia of Integer Sequences." Spanier, J.

Cody.) Similarly, we also implement special-case code for real z, where the imaginary part of w is Dawson's integral. p.297. In order of increasing accuracy, they are: erf ⁡ ( x ) ≈ 1 − 1 ( 1 + a 1 x + a 2 x 2 + a 3 x Error Function Properties Referenced on Wolfram|Alpha: Erf CITE THIS AS: Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource.

Gamma: Exploring Euler's Constant. History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Despite the name "imaginary error function", erfi ⁡ ( x ) {\displaystyle \operatorname ⁡ 8 (x)} is real when x is real. navigate here Wolfram Language» Knowledge-based programming for everyone.

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. Similarly, the erf, erfc, erfcx, erfi, and Dawson functions are computed by calling: extern std::complex Faddeeva::erf(std::complex z, double relerr=0); extern std::complex Faddeeva::erfc(std::complex z, double relerr=0); extern std::complex Faddeeva::erfcx(std::complex z, double relerr=0); 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 For iterative calculation of the above series, the following alternative formulation may be useful: erf ⁡ ( z ) = 2 π ∑ n = 0 ∞ ( z ∏ k

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 See also Related functions Gaussian integral, over the whole real line Gaussian function, derivative Dawson function, renormalized imaginary error function Goodwin–Staton integral In probability Normal distribution Normal cumulative distribution function, a Intermediate levels of Im(ƒ)=constant are shown with thin green lines. 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.

in the vicinity of |z|=1 [although comparison with other compilers suggests that this may be a problem specific to gfortran]. 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. For erfi and the Dawson function, there are simplifications that occur for real x as noted. Thus π 1 / 2 H ( y ) = Im ⁡ ∫ 0 ∞ d k exp ⁡ [ − k 2 / 4 + i k y ] {\displaystyle