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Date: 22-5-2019
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The inverse erf function is the inverse function of the erf function such that
(1) |
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(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) |
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(5) |
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(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) |
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(12) |
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(13) |
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(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."
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