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Inverse Erfc

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InverseErfc

The inverse erf function is the inverse function erfc^(-1)(z) of erfc(x) such that

erfc(erfc^(-1)(x))=erfc^(-1)(erfc(x)),
(1)

with the first identity holding for 0<x<2 and the second for x in R. It is implemented in the Wolfram Language as InverseErfc[z].

It is related to inverse erf by

erfc^(-1)(1-x)=erf^(-1)(x).
(2)

It has the special values

erfc^(-1)(0)=infty
(3)
erfc^(-1)(1)=0
(4)
erfc^(-1)(2)=-infty.
(5)

It has the derivative

d/(dx)erfc^(-1)(x)=-1/2sqrt(pi)e^([erfc^(-1)(x)]^2),
(6)

and its indefinite integral is

interfc^(-1)(x)dx=(e^(-[erfc^(-1)(x)]^2))/(sqrt(pi))
(7)

(which follows from the method of Parker 1955).

The Taylor series about 1 is given by

erfc^(-1)(x-1)=-1/2sqrt(pi)(x-1)-1/(24)pi^(3/2)(x-1)^3-7/(960)pi^(5/2)(x-1)^5-(127)/(80640)pi^(7/2)(x-1)^7-...
(8)

(OEIS A002067 and A007019).

See also

Erfc, Inverse Erf

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/InverseErfc/

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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/M3126, A092676, and A092677 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Inverse Erfc." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseErfc.html

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