Generalized Error Function
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where it was expressed through the following integral: Later C. Kramp (1799) used this integral for the definition of the complementary error function . P.‐S. Laplace (1812) derived an asymptotic error function integral expansion of the error function. The probability integrals were so named because error function calculator they are widely applied in the theory of probability, in both normal and limit distributions. To obtain, say, a error function table normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed. The inverse was systematically investigated in the second half of the twentieth century, error function matlab especially by J. R. Philip (1960) and A. J. 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 inverse error function , the inverse of the generalized error function , and the inverse complementary error function are defined through the following formulas:Inverse Error Function
These seven functions are typically called probability integrals and their inverses. 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 look at the graphics for the probability integrals and inverses along the real axis. Connections within the group of probability integrals and inverses and with other function groups Representations through more general functions The probability integrals , , , and are the particular cases of two more general functions: hypergeometric and Meijer G functions. For example, they can be represented through the confluent hypergeometric functions and : Representations of the probability integrals , , , and through classical Meijer G functions are rather simple: The factor in the last four formulas can be removed by changing the classical Meijer G functions to the generalized one: The probability integrals , , , and are the particular cases of the incomplete gamma function, regularized incomplete gamma function, and exponential integral : Representations through related equivalent functions The probability integrals , , and can be represented through Fresnel integrals by(5 formulas) http://functions.wolfram.com/GammaBetaErf/Erf2/ Integral representations (1 formula) Differential equations (4 formulas) Differentiation (12 formulas) Integration (4 formulas) Representations through more general functions (11 formulas) Representations through equivalent functions (1 formula) Zeros (1 formula) Erf[z]
Keywords: Dawson’s integral, error functions Permalink: error function http://dlmf.nist.gov/7.16 See also: info for 7 Generalizations of the error function and Dawson’s integral are ∫0xe-tpdt error function table and ∫0xetpdt. These functions can be expressed in terms of the incomplete gamma function γ(a,z) (§8.2(i)) by change of integration variable. © 2010–2016 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.13; Release date 2016-09-16. A printed companion is available. 7.15 Sums7.17 Inverse Error Functions
error functions with applications to probability and heat conductionArticle · January 2002 with 19 Reads1st M. A. Chaudhry2nd Asghar Qadir3rd Syed M. Zubair42.13 · King Fahd University of Petroleum and MineralsAbstractA generalization of the classical error functions is introduced. Applications of these functions in probability theory and heat conduction problems are discussed. A graphical and tabular representation of the functions is presented.Do you want to read the rest of this article?Request full-text CitationsCitations6ReferencesReferences0Certain results on $q$-starlike and $q$-convex error functions"Its inverse, introduced by Carlitz [8], which we will denote by inverf , appears in multiple areas of mathematics and the natural sciences. A few examples include concentration-dependent diffusion problems [31], solutions to Einstein's scalar-field equations and in heat conduction problem [9, 27]. Now, we recall the definitions of fundamental class of analytic functions. "[Show abstract] [Hide abstract] ABSTRACT: The error function occurs widely in multiple areas of mathematics, mathematical physics and natural sciences. There has been no work in this area for the past four decades. In this article, we estimate the coefficient bounds with q-difference operator for certain classes of the spirallike starlike and convex error function associated with convolution product using subordination as well as quasi-subordination. Though this concept is an untrodden path in the field of complex function theory, it will prove to be an encouraging future study for researchers on error function.Article · Jul 2016 · Applied Mathematics and ComputationS. KanasC. RamachandranL. VanithaReadThe generalization of special functions[Show abstract] [Hide abstract] ABSTRACT: The question of what special functions should be taken to be is addressed. Using examples of generalizations that have proved useful, with special reference to those develo