On The Error Function Of A Complex Argument
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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: Wed Oct 19 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus complementary error function and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... error function calculator History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution error function table (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 1990, p.341) define without the leading factor
Inverse Error Function
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 function. Erf can also be defined as a Maclaurin series (6) error function matlab (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 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp.8-9). Definite integrals involving
Issue 1, pp 33–40On the error function of a complex argumentAuthorsAuthors and affiliationsJoseph KestinLeif N. PersenArticleReceived: 10 February 1955DOI: 10.1007/BF01600725Cite this article as: Kestin, J. & Persen, L.N. Journal of Applied Mathematics and Physics (ZAMP) (1956) erf(inf) 7: 33. doi:10.1007/BF01600725 2 Citations 220 Views ZusammenfassungIn der Einleitung werden die wichtigsten
Error Function Excel
bisherigen Arbeiten über die Fehlerfunktion mit komplexem Argument kurz besprochen. Die der vorliegenden Untersuchung unterworfene FunktionK(z) ist in Gleichung
Error Function Python
(1) definiert. Mit Hilfe der Laplace-Transformation und ihrer Umkehrformel wird gezeigt, wie sich die zwei Identitäten in Gleichung (3) behandeln lassen, so dass sowohl der Real-als auch der Imaginärteil der FunktionK(z) sich http://mathworld.wolfram.com/Erf.html in zwei Teile aufspalten lässt, was aus den Gleichungen (9) und (9a) hervorgeht. Der erste Teil kann durch elementare Funktionen ausgedrückt werden, während der zweite Teil sich durch zwei Integrale darstellen lässt. Für die praktische Anwendung der angegebenen Ausdrücke sind in den Gleichungen (12), (12a) und (14a), (14b) die Ausdrücke in kartesischen bzw. polaren Koordinaten umgeschrieben worden. Der Vorteil der angegebenen Aufspaltung liegt darin, http://link.springer.com/article/10.1007/BF01600725 dass die in den Ausdrücken auftretenden Integrale monoton abnehmende Funktionen der unabhängigen Veränderlichen (x, y) darstellen und sich deswegen leicht numerisch ausrechnen lassen. Der Schwingungsanteil der FunktionK(z) ist ausschliesslich durch elementare Funktionen ausgedrückt. Im Appendix ist der Rechnungsvorgang, der zu den angegebenen Ausdrücken führt, näher umschrieben.References[1] J. B. Rosser,Theory and Application of \(\mathop \smallint \limits_0^z e^{ - x^2 } dxand\mathop \smallint \limits_0^z e^{ - p^2 y^2 } dy\mathop \smallint \limits_0^y e^{ - x^2 } dx\), Rep. OSRD 5861 prepared by Allegany Ballistics Laboratory, Contr. OEMsr-273 1945.[2] P. C. Clemmow andC. M. Munford, A Table of\(\sqrt {\left( {\frac{1}{2}\pi } \right)} e^{\frac{1}{2}i\pi x^2 } \mathop \smallint \limits_\varrho ^\infty e^{ - \frac{1}{2}i\pi x^2 } dx\) for Complex Values of x, Phil. Trans. [A]245, 189 (1952).Google Scholar[3] W. Skwirzynski,Evaluation of erfc (z), Marconi's Wireless Telegraph Co., Res. Division, Rep. RD. 992, Great Baddow, Essex, 1952 (Unpublished).Google Scholar[4] W. F. Cahill,A Short Table of the Error Function of Complex Arguments, NBS Project 1102-10-1104, Report 3034, Washington, 1953.[5] T. Laible,Höhenkarte des Fehlerintegrals, ZAMP2, 484 (1951).Google Scholar[6] J. Kestin andL. N. Persen,Slow Oscillations of an Infinite Plate and an Infinite Disk in a Viscous Fluid, Brown University Repo
Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation Home Symbolic Math Toolbox Examples Functions and Other Reference Release https://www.mathworks.com/help/symbolic/mupad_ref/erfi.html Notes PDF Documentation MuPAD Mathematics Mathematical Constants and Functions Special Functions Error and Exponential Integral Functions Symbolic Math Toolbox MuPAD Functions erfi On this page Syntax Description Environment Interactions Examples Example 1 Example 2 Example 3 Parameters Return Values Algorithms See Also More About This is machine translation Translated by Mouse over text to see original. Click the button below to return to error function the English verison of the page. Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian Italian Japanese Korean Latvian Lithuanian Malay Maltese Norwegian Polish Portuguese Romanian Russian Slovak Slovenian Spanish Swedish Thai Turkish Ukrainian Vietnamese Welsh MathWorks Machine Translation The automated translation on the error of this page is provided by a general purpose third party translator tool. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. Translate erfiImaginary error functionexpand all in page MuPAD notebooks are not recommended. Use MATLAB live scripts instead.MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.Syntaxerfi(x) Descriptionerfi(x)=−ierf(ix)=2π∫0xet2dt computes the imaginary error function.This function is defined for all complex arguments x. For floating-point arguments, erfi returns floating-point results. The implemented exact values are: erfi(0) = 0, erfi(∞) = ∞, erfi(-∞) = -∞, erfi(i∞) = i, and erfi(-i∞) = -i. For all other arguments, the error function returns symbolic function calls.For the function call erfi(x) = -i*erf(i*x) = i*(erfc(i*x) - 1) with floating-point arguments of large absolute value, internal numerical underflow or overflow can happen. If a call to erfc causes underflow or overflow, this function returns:The result truncated to 0.0 if x is a large positive real numberThe result rounded to 2.0 if x is a large negative real numberRD_NAN if x is a l
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