Exponential Error Function
Contents |
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 derivative of error function = 2 π ∫ 0 x e − t 2 d t . erf function calculator {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt error function table {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf
Inverse Error Function
( x ) = 2 π ∫ x ∞ 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 error function matlab be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( 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.
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 nurturedbyEricWeisstein at WolframResearch Calculus and error function properties Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and
Error Function Excel
Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is
Error Function Python
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 of . https://en.wikipedia.org/wiki/Error_function 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) (7) (OEIS A007680). http://mathworld.wolfram.com/Erf.html 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 include Definite integrals involving include (34) (35)
Popular Topics 5.2a 5.2 5.1 5.0 4.4 4.3b 4.3a 4.3 4.2a 4.2 4.1 4.0a 4.0 3.5a AC/DC ModuleAcoustics ModuleCFDChemical Reaction EngineeringDefinitions and operatorsFluid FlowGeometryHeat TransferLiveLink productsMaterials and functionsMEMSMeshMicrofluidicsPhysicsResults and visualizationRF ModuleSolvingStructural Mechanics List all http://www.comsol.com/community/forums/general/thread/16250/ discussions Share this pageEmailLinkedInFacebookTwitterDeliciousDiggStumbleuponMore services... exponential function error Topics: 4.0a Thread index | Previous thread | Next thread | Start a new discussion RSS feed | Turn on email notifications | 11 Replies Last post: November 16, 2011 12:35pm UTC JiYoung Park March 31, 2011 12:09pm UTC exponential function error Hello~ In global definition>funtions I defined the function : error function exp(-1/(T-Tg)) , T was arguments. At T=Tg, the exponential value should be zero but it was infinity. can anybody help?? Reply | Reply with Quote | Send private message | Report Abuse Amir Fadel March 31, 2011 12:21pm UTC in response to JiYoung Park Re: exponential function error Hi, If you approach Tg from the left, i.e. is T is almost Tg but slightly less, exponential error function then the function goes to +infinity (just try to solve the limit, the exponent wil go to +infinity), if you approach Tg from the right then it goes to zero (the exponent goes to -infinity) at the limit never really reaching zero. Hello~ In global definition>funtions I defined the function : exp(-1/(T-Tg)) , T was arguments. At T=Tg, the exponential value should be zero but it was infinity. can anybody help?? Reply | Reply with Quote | Send private message | Report Abuse JiYoung Park March 31, 2011 12:28pm UTC in response to Amir Fadel Re: exponential function error Dear am fa thank you for replying. I understood your comment but, how can i treat that function to be zero? Is there any limit operator in comsol? Reply | Reply with Quote | Send private message | Report Abuse JiYoung Park March 31, 2011 12:36pm UTC in response to JiYoung Park Re: exponential function error I tried with : exp(1/(Tg-T)) but the answer was infinity... Reply | Reply with Quote | Send private message | Report Abuse Magnus Ringh COMSOL EmployeeSweden March 31, 2011 1:51pm UTC in response to JiYoung Park Re: ex
be down. Please try the request again. Your cache administrator is webmaster. Generated Sat, 15 Oct 2016 11:35:34 GMT by s_ac15 (squid/3.5.20)