Error Function Integrate
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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 = 2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin − gamma function integral 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − normal distribution integral 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc
Gaussian Integral
( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1
Error Function Integral Table
(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 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 / error function integral calculation 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. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{-z^ 6}\operatorname 5 (-iz)=\operatorname 4 (-iz).} Contents 1 The name "error function" 2 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann series 2.4 Inverse functions 2.5 Asymptotic expansion 2.6 Continued fraction expansion 2.7 Integral of error function with G
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Error Function Values
& Physiology Astronomy Astrophysics beta Biology Chemistry Earth Science Environmental Science erfc function beta Organic Chemistry Physics Math Algebra Calculus Geometry Prealgebra Precalculus Statistics Trigonometry Social Science Psychology beta Humanities erfc integral English Grammar U.S. History beta World History beta ... and beyond What's Next Socratic Meta Scratchpad Questions Topics × × Get our new iOS app! Snap a picture https://en.wikipedia.org/wiki/Error_function of your homework & find answers, explanations and videos Get the App or go to Socratic.org/ios on your iPhone Enter your phone number and we'll send you a download link Text me or go to Socratic.org/ios on your iPhone What is the integral of the error function? Calculus Introduction to Integration Definite and indefinite integrals 1 https://socratic.org/questions/what-is-the-integral-of-the-error-function Answer Write your answer here... Start with a one sentence answer Then teach the underlying concepts Don't copy without citing sources How to add symbols & How to write great answers preview ? Answer Write a one sentence answer... Answer: Explanation Explain in detail... Explanation: I want someone to double check my answer Describe your changes (optional) 200 Cancel Update answer 2 sente Share Feb 20, 2016 Answer: #int"erf"(x)dx = x"erf"(x)+e^(-x^2)/sqrt(pi)+C# Explanation: We will use the definition of the error function: #"erf"(x) = 2/sqrt(pi)int_0^xe^(-t^2)dt# Along with integration by substitution, integration by parts, and the fundamental theorem of calculus. Integration by Parts: Let #u = "erf"(x)# and #dv = dt# Then, by the fundamental theorem of calculus, #du = 2/sqrt(pi)e^(-x^2)# and #v = x# By the integration by parts formula #intudv = uv - intvdu# #int"erf"(x)dx = x"erf"(x) - int2/sqrt(pi)xe^(-x^2)dx# Integration by Substitution: To evaluate the remaining integral, let #u = -x^2# Then #du = -2xdx# and so #-int2/sqrt(pi)xe^(-x^2)dx = 1/sqrt(pi)inte^udu# #=1/sqrt(pi)e^u + C# #=e^(-x^2)/sqrt(pi)+C# Putting it al
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies http://math.stackexchange.com/questions/37889/why-is-the-error-function-defined-as-it-is 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 professionals in related fields. Join them; it only takes a minute: Sign error function 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 Why is the error function defined as it is? up vote 35 down vote favorite 6 $\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x error function integral e^{-t^2}dt.$$ Of course, it is closely related to the normal cdf $$\Phi(x) = P(N < x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt$$ (where $N \sim N(0,1)$ is a standard normal) by the expression $\erf(x) = 2\Phi(x \sqrt{2})-1$. My question is: Why is it natural or useful to define $\erf$ normalized in this way? I may be biased: as a probabilist, I think much more naturally in terms of $\Phi$. However, anytime I want to compute something, I find that my calculator or math library only provides $\erf$, and I have to go check a textbook or Wikipedia to remember where all the $1$s and $2$s go. Being charitable, I have to assume that $\erf$ was invented for some reason other than to cause me annoyance, so I would like to know what it is. If nothing else, it might help me remember the definition. Wikipedia says: The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics. So perhaps a practitioner of one of these mysterious "other branches of mathematics"
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