Error Function Origin
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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 − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x http://www.originlab.com/doc/LabTalk/ref/Erf-func ) = 1 − erf ( 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 https://en.wikipedia.org/wiki/Error_function 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 / 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 "imaginar
all the time, but I have no idea how it got its name.UpdateCancelAnswer Wiki1 Answer Arun Iyer, I dabble with mathematics, not really good at it yetWritten 167w https://www.quora.com/What-is-the-origin-of-the-name-error-function agoFrom Earliest Known Uses of Some of the Words of Mathematics (E):Error function. In the course of the 19th century the function from the theory of errors appeared in several contexts unrelated to probability, e.g. refraction and heat conduction. In 1871 J. W. Glaisher wrote that "Erf(x) may fairly claim at present to rank in importance next to the trigonometrical and logarithmic functions." Glaisher introduced error function the symbol Erf and the name error function for a particular form of the law as follows:As it is necessary that the function should have a name, and as I do not know that any has been suggested, I propose to call it the Error-function ... and to write ("On a Class of Definite Integrals, Philosophical Magazine, 42, 1871, p. 296) Glaisher also introduced the error function origin error-function-complement with the symbol Erfc. The names and symbols continue to be used though often the definitions are slightly different from Glaisher's.When R. A. Fisher wrote about regression and the analysis of variance he was reviving the theory of errors. He and his followers reinterpreted old terms and produced new ones. They considered their extensions so radical that they were no longer working in the theory of errors; that was now part of history: see Fisher's Statistical Methods for Research Workers (1925, §1).2.4k Views · View UpvotesView More AnswersRelated QuestionsWhat makes an 'elementary' function different from something like the error function or Lambert's W function?When does a taylor series converge to the original function?Why are functions in mathematics called functions? Who first came up with that name anyway?Can the name of a mathematical function have multiple letters?Why are odd functions described as being "symmetric about the origin"?What is the origin story of e (base of natural exponential function)?What is the formulation of this expression: "shrinkage towards the origin", using a mathematical function?What is the origin of the names of the different Algebraic Structures?The error term in third order taylor polynomial approximation of the function