Inverse Erf

The 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 . 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 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 .

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."