Limits Involving Error Function
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Integral Of Error Function
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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 inverse error function and 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 How to compute this limit involving complementary error functions up vote 1 down vote favorite I am trying error function matlab to take the following limit $$\lim_{x\to \infty } \, \frac{2 x \operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]}{t \operatorname{erfc}\left[-\frac{x}{\sqrt{2}}\right]}$$ my first thoughts were to use LHospital's rule after making the top complementary error function as the denominator of the denominator. Upon differentiating I got the following $$\lim_{x\to \infty } \, \frac{\sqrt{2 \pi } \exp\left[\frac{\left(t^2+1\right) x^2}{2 t^2}\right] \operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]^2}{\exp\left[\frac{x^2}{2}\right] \left(\operatorname{erf}\left[\frac{x}{\sqrt{2}}\right]+1\right)+t \exp\left[\frac{x^2}{2 t^2} \right]\operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]}$$ which seems even more complicated. Maybe using an identity or some form of expansion for the complementary error function may lead to a simpler expression. Any help will be greatly appreciated. Thank you. calculus limits error-function share|cite|improve this question edited Sep 6 at 20:42 asked Sep 6 at 20:03 Comic Book Guy 1,96731946 add a comment| 3 Answers 3 active oldest votes up vote 1 down vote accepted If you look here and here, you will see the asymptotic expansion $$\operatorname{erfc}(z) = \frac{e^{-z^2}}{z\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n - 1)!!}{(2z^2)^n}$$ $$\operatorname{erfc}(-z) = 2-\frac{e^{-z^2}}{z\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n - 1)!!}{(2z^2)^n}$$So, for very large $z$, keeping the first term only makes $$
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Error Function Properties
Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> error function excel Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian
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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 . Erf is implemented in the Wolfram Language as Erf[z]. http://math.stackexchange.com/questions/1917071/how-to-compute-this-limit-involving-complementary-error-functions 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). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) http://mathworld.wolfram.com/Erf.html (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) (36) (37) (38) The first two of these appear in Prudnikov et al. (1990, p.123, eqns. 2.8.19.8 and 2.8.19.11), with , . A complex gen
Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation https://www.mathworks.com/help/symbolic/mupad_ref/erf.html Home Symbolic Math Toolbox Examples Functions and Other Reference Release Notes PDF Documentation MuPAD Mathematics Mathematical Constants and Functions Special Functions Error and Exponential Integral Functions Symbolic Math Toolbox MuPAD Functions erf On this page Syntax Description Environment Interactions Examples Example 1 Example 2 Example 3 Parameters Return Values Algorithms See Also More About error function This is machine translation Translated by Mouse over text to see original. Click the button below to return to 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 limits involving error 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 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 erfError 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.Syntaxerf(x) Descriptionerf(x) represents the error function 2π∫0xe−t2dt.This function is defined for all complex arguments x. For floating-point arguments, erf returns floating-point results. The implemented exact values are: erf(0) = 0, erf(∞) = 1, erf(-∞) = -1, erf(i ∞) = i ∞, and erf(-i ∞) = -i ∞. For all other arguments, the error function returns symbolic function calls.For the function call erf(x) = 1 - erfc(x) wit