Error Function Approximation With Elementary Functions
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ error function approximation formula 0 x e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5
Gamma Function Approximation
(x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ approximation q function 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ error function values e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc
Normal Distribution Approximation
( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. 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 fun
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 Created, developed, and gaussian approximation nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> complementary error function Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered
Error Function Calculator
in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by (1) Note that some authors (e.g., Whittaker and Watson https://en.wikipedia.org/wiki/Error_function 1990, p.341) define without the leading factor of . Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For , (5) where is the incomplete gamma http://mathworld.wolfram.com/Erf.html function. Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). 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). Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as (25) (26) Its derivative is (27) where is a Hermite polynomial. The first derivative is (28) and the integral is (29) Min Max Re Im Erf can also be extended to the complex plane, as illustrated above. A simple integral involving erf that Wolfram Language cannot do is given by (30) (M.R.D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include (31) (M.R.D'Orsogna, pers. comm., Dec.15, 2005). Erf has the continued fraction (32) (33) (Wall 1948, p.357), first stated by Laplace in
von GoogleAnmeldenAusgeblendete FelderBooksbooks.google.de - This book constitutes the thoroughly refereed post-proceedings of the Dagstuhl Seminar https://books.google.com/books?id=5g5tCQAAQBAJ&pg=PA35&lpg=PA35&dq=error+function+approximation+with+elementary+functions&source=bl&ots=5S07qeb1Mg&sig=JW0x9S8A_w0xxMZO32lpG-7FwkU&hl=en&sa=X&ved=0ahUKEwjxh8io98vPAhXh64MK 08021 on Numerical Validation in Current Hardware Architectures held at Dagstuhl Castle, Germany, in January 2008. The 16 revised full papers presented were selected during two rounds of reviewing and improvements....https://books.google.de/books/about/Numerical_Validation_in_Current_Hardware.html?hl=de&id=5g5tCQAAQBAJ&utm_source=gb-gplus-shareNumerical Validation in Current Hardware ArchitecturesMeine BücherHilfeErweiterte BuchsucheE-Book kaufen - error function 49,97 €Nach Druckexemplar suchenSpringer ShopAmazon.deBuch.de - €71,39Buchkatalog.deLibri.deWeltbild.deIn Bücherei suchenAlle Händler»Numerical Validation in Current Hardware Architectures: International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 6-11, 2008, Revised PapersAnnie A.M. Cuyt, Walter Krämer, Wolfram Luther, Peter MarksteinSpringer, 28.04.2009 - 263 Seiten 0 Rezensionenhttps://books.google.de/books/about/Numerical_Validation_in_Current_Hardware.html?hl=de&id=5g5tCQAAQBAJThis error function approximation book constitutes the thoroughly refereed post-proceedings of the Dagstuhl Seminar 08021 on Numerical Validation in Current Hardware Architectures held at Dagstuhl Castle, Germany, in January 2008. The 16 revised full papers presented were selected during two rounds of reviewing and improvements. The papers are organized in topical sections on languages, software systems and tools, new verification techniques based on interval arithmetic, applications in science and engineering, and novel approaches to verification. Voransicht des Buches » Was andere dazu sagen-Rezension schreibenEs wurden keine Rezensionen gefunden.Ausgewählte SeitenTitelseiteInhaltsverzeichnisIndexVerweiseAndere Ausgaben - Alle anzeigenNumerical Validation in Current Hardware Architectures: International ...Annie A.M. Cuyt,Walter Krämer,Wolfram LutherEingeschränkte Leseprobe - 2009Häufige Begriffe und Wortgruppenaccuracy algorithm application approximation arithmetic operations automatic differentiation basic C-XSC CELL proces