The inverse erf function is the inverse function of the erf function such that
(1)
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(2)
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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)
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It has the special values
(4)
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(5)
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(6)
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It is apparently not known if
(7)
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(OEIS A069286) can be written in closed form.
It satisfies the equation
(8)
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where is the inverse erfc function.
It has the derivative
(9)
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and its integral is
(10)
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(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)
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(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)
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(OEIS A002067 and A007019). Written in simplified form so that the coefficient of is 1,
(16)
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(OEIS A092676 and A092677). The th coefficient of this series can be computed as
(17)
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where is given by the recurrence equation
(18)
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with initial condition .