Inverse Error Function Wolfram Alpha
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Inverse Error Function Excel
Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Calculus>Integrals>Definite Integrals> History and Terminology>Wolfram Language Commands> Inverse Erf The inverse error function calculator inverse erf function is the inverse function of the erf function such that (1) (2) with the first identity holding for and the second for . inverse error function matlab It is implemented in the Wolfram Language as InverseErf[x]. It is an odd function since (3) It has the special values (4) (5) (6) It is apparently not known if (7) (OEIS A069286) can be written in closed form. It satisfies the equation (8) where is the inverse erfc function. It has the
Inverse Complementary Error Function
derivative (9) and its integral is (10) (which follows from the method of Parker 1955). Definite integrals are given by (11) (12) (13) (14) (OEIS A087197 and A114864), where is the Euler-Mascheroni constant and is the natural logarithm of 2. The Maclaurin series of is given by (15) (OEIS A002067 and A007019). Written in simplified form so that the coefficient of is 1, (16) (OEIS A092676 and A092677). The th coefficient of this series can be computed as (17) where is given by the recurrence equation (18) with initial condition . SEE ALSO: Confidence Interval, Erf, Inverse Erfc, Probable Error RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/InverseErf/, http://functions.wolfram.com/GammaBetaErf/InverseErf2/ REFERENCES: Bergeron, F.; Labelle, G.; and Leroux, P. Ch.5 in Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, 1998. Carlitz, L. "The Inverse of the Error Function." Pacific J. Math. 13, 459-470, 1963. Parker, F.D. "Integrals of Inverse Functions." Amer. Math. Monthly 62, 439-440, 1955. Sloane, N.J.A. Sequences A002067/M4458, A007019/M
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Erf Function Calculator
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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 = https://en.wikipedia.org/wiki/Error_function 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 ) = 1 − erf ( x error function ) = 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 defines erfcx, the scaled complementary error function[3] (which can be used instead inverse error function 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 "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. When the error function is ev