Derivative Of The Error Function
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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 derivative complementary error function entries Last updated: Tue Sep 27 2016 Created, developed, and differentiation error function nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite
Derivative Q Function
Integrals> More... Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function
Derivative Gamma Function
defined by (1) (2) It is implemented in the Wolfram Language as Erfc[z]. Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . For , (3) where is the incomplete gamma function. The derivative is given by (4) and the indefinite integral by (5) derivative normal distribution It has the special values (6) (7) (8) It satisfies the identity (9) It has definite integrals (10) (11) (12) For , is bounded by (13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above. A generalization is obtained from the erfc differential equation (14) (Abramowitz and Stegun 1972, p.299; Zwillinger 1997, p.122). The general solution is then (15) where is the repeated erfc integral. For integer , (16) (17) (18) (19) (Abramowitz and Stegun 1972, p.299), where is a confluent hypergeometric function of the first kind and is a gamma function. The first few values, extended by the definition for and 0, are given by (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). "Repeated Integrals of the Error Function." §7.2 in Handbook of Mathematical Functions wi
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Derivative Gaussian
site About Us Learn more about Stack Overflow the company Business Learn error function values more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics differentiation of complementary error function Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how http://mathworld.wolfram.com/Erfc.html it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top derivative of error function up vote 0 down vote favorite How can I calculate the derivatives $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \mu}$$ and $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \sigma}$$ where $\mbox{erf}$ denotes the error function can be given by $$\mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp(-t^2)\,dt$$ I have tried it using WA derivative calculator http://math.stackexchange.com/questions/1755149/derivative-of-error-function but I am not able to understand the steps. derivatives error-function share|cite|improve this question edited Apr 23 at 9:02 kamil09875 4,3592729 asked Apr 23 at 7:44 Rakesh 11 The error function erf($x$) is just $\frac{2}{\sqrt\pi}\int_0^xe^{-t^2}\ dt$, so its derivative is just $\frac{2}{\sqrt\pi}e^{-x^2}$. All you have to do for your examples is use the chain rule. –almagest Apr 23 at 7:58 add a comment| 1 Answer 1 active oldest votes up vote 0 down vote You have error in your definition of error function :-). The definition of error function is $$\operatorname{erf}(x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\mathrm dt = \int_0^x \frac{2}{\sqrt\pi}e^{-t^2}\,\mathrm dt.$$ Derivative of this integral with variable is it's integrand applied to upper boundary and multiplicated by boundary's derivative. ($\frac{\partial x}{\partial x}=1$) $$\frac{\partial \operatorname{erf}(x) }{\partial x}=1\cdot\frac{2}{\sqrt\pi}e^{-x^2}$$ The next step is calculating derivative of a composite function. I hope you can do it yourself. ==Added== You should treat $t$ and $\mu$ as a parameters. For example: $$\frac{\partial \frac{\ln(t)-\mu}{\sqrt{2}\sigma}}{\partial \sigma}=\frac{\ln(t)-\mu}{\sqrt{2}}{\ln|\sigma|}$$ Continue it. share|cite|improve this answer edited Apr 23 at 8:20 answered Apr 23 at 7:59 Georgy Dunaev 1588 Thanks Georgy Dunaev It was a typographical er
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more http://math.stackexchange.com/questions/712434/erfaib-error-function-separate-into-real-and-imaginary-part about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask https://www.youtube.com/watch?v=CcFUQhorgdc Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level 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 erf(a+ib) error function separate into real error function and imaginary part up vote 5 down vote favorite 3 Is there an 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; complementary error function 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 $$\left |\epsilon(x,y) \right | \le 10^{-16} |\operatorname*{erf}{(x+i y)}| $$ This accuracy is valid for all $x$ and $y$, i.e., the complex plane. I will present a derivation of this result to show you where the error term comes from. Consider the definition of the error function in the complex plane: $$\operatorname*{erf}{z} = \frac{2}{\sqrt{\pi}} \int_{\Gamma} d\zeta \, e^{-\zeta^2}$$ where $\Gamma$ is any path in the complex plane from $\zeta = 0$ to $\zeta=z$. Consider, then, the special case where $\Gamma$ is the path that runs from $0$ to $x$ along the real axis, then from $x$ to $z=x+i y$ parallel to the imaginary axis. Seen this way, the error function of a complex number is equal to $$\operatorname*{erf}{(x+i y)} = \operatorname*{erf}{x} + i \frac{2}{\sqrt{\pi}} e^{-x^2} \int_0^y du \, e^{u^2} \cos{2 x u} \\ + \frac{2}{
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