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Inverse Erf


InverseErf

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

erf(erf^(-1)(x))=x
(1)
erf^(-1)(erf(x))=x,
(2)

with the first identity holding for -1<x<1 and the second for x in R. It is implemented in the Wolfram Language as InverseErf[x].

It is an odd function since

 erf^(-1)(x)=-erf^(-1)(-x).
(3)

It has the special values

erf^(-1)(-1)=-infty
(4)
erf^(-1)(0)=0
(5)
erf^(-1)(1)=infty.
(6)

It is apparently not known if

 erf^(-1)(1/2)=0.47693627...
(7)

(OEIS A069286) can be written in closed form.

It satisfies the equation

 erf^(-1)(x)=erfc^(-1)(1-x)
(8)

where erfc^(-1)(x) is the inverse erfc function.

It has the derivative

 d/(dx)erf^(-1)(x)=1/2sqrt(pi)e^([erf^(-1)(x)]^2),
(9)

and its integral is

 interf^(-1)(x)dx=-(e^(-[erf^(-1)(x)]^2))/(sqrt(pi))
(10)

(which follows from the method of Parker 1955).

Definite integrals are given by

int_0^1erf^(-1)(z)dz=1/(sqrt(pi))
(11)
=0.564189582...
(12)
int_0^1ln[erf^(-1)(z)]dz=-(1/2gamma+ln2)
(13)
=-0.98175...
(14)

(OEIS A087197 and A114864), where gamma is the Euler-Mascheroni constant and ln2 is the natural logarithm of 2.

The Maclaurin series of erf^(-1)(x) is given by

 erf^(-1)(x)=sqrt(pi)(1/2x+1/(24)pix^3+7/(960)pi^2x^5+(127)/(80640)pi^3x^7+...)
(15)

(OEIS A002067 and A007019). Written in simplified form so that the coefficient of x is 1,

 erf^(-1)((2x)/(sqrt(pi)))=x+1/3x^3+7/(30)x^5+(127)/(630)x^7+...
(16)

(OEIS A092676 and A092677). The nth coefficient of this series can be computed as

 a_n=(c_n)/(2n+1),
(17)

where c_n is given by the recurrence equation

 c_n=sum_(k=0)^(n-1)(c_kc_(n-1-k))/((k+1)(2k+1))
(18)

with initial condition c_0=1.


See also

Confidence Interval, Erf, Inverse Erfc, Probable Error

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/InverseErf/, http://functions.wolfram.com/GammaBetaErf/InverseErf2/

Explore with Wolfram|Alpha

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, A069286, A087197, A092676, A092677, A114859, A114860, and A114864 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

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

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