Complex Error Function Properties
Contents |
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 complex error function matlab at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive error function of complex argument Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in
Complex Gamma Function
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 1990,
Q Function Properties
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 function. Erf error function values 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 1805 and Legendre in 1826
here for a quick overview of the site Help Center Detailed answers to any questions you might have http://math.stackexchange.com/questions/712434/erfaib-error-function-separate-into-real-and-imaginary-part Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and error function professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top erf(a+ib) error function separate into real and imaginary part up vote 5 down vote favorite 3 Is there an complex error function easy way to separate erf(a+ib) into real and imaginary part? calculus integration complex-analysis contour-integration share|cite|improve this question edited Mar 14 '14 at 22:49 Ron Gordon 109k12130221 asked Mar 14 '14 at 19:04 Sleepyhead 1385 add a comment| 3 Answers 3 active oldest votes up vote 6 down vote I'm not sure if you are interested in an analytical answer or a computational answer; these are two different things. The analytical answer is...not really, unless you consider GEdgar's answer useful. (And one might.) The computational answer is a resounding yes. A result found in Abramowitz & Stegun claims the following: $$\operatorname*{erf}(x+i y) = \operatorname*{erf}{x} + \frac{e^{-x^2}}{2 \pi x} [(1-\cos{2 x y})+i \sin{2 x y}]\\ + \frac{2}{\pi} e^{-x^2} \sum_{k=1}^{\infty} \frac{e^{-k^2/4}}{k^2+4 x^2}[f_k(x,y)+i g_k(x,y)] + \epsilon(x,y) $$ where $$f_k(x,y) = 2 x (1-\cos{2 x y} \cosh{ k y}) + k\sin{2 x y} \sinh{k y}$$ $$g_k(x,y) = 2 x \sin{2 x y} \cosh{k y} + k\cos{2 x y} \sinh{k y}$$ Then $$\left |\epsilon(x,y) \right | \le 10^{-16} |\operatorname*{erf}{(
be down. Please try the request again. Your cache administrator is webmaster. Generated Wed, 05 Oct 2016 23:54:01 GMT by s_hv978 (squid/3.5.20)