Inverse Erfc
المؤلف:
Bergeron, F.; Labelle, G.; and Leroux, P
المصدر:
Ch. 5 in Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, 1998.
الجزء والصفحة:
...
28-4-2019
1955
Inverse Erfc
The inverse erf function is the inverse function 
of 
such that
 |
(1)
|
with the first identity holding for 
and the second for 
. It is implemented in the Wolfram Language as InverseErfc[z].
It is related to inverse erf by
 |
(2)
|
It has the special values
It has the derivative
![d/(dx)erfc^(-1)(x)=-1/2sqrt(pi)e^([erfc^(-1)(x)]^2),](http://mathworld.wolfram.com/images/equations/InverseErfc/NumberedEquation3.gif) |
(6)
|
and its indefinite integral is
![interfc^(-1)(x)dx=(e^(-[erfc^(-1)(x)]^2))/(sqrt(pi))](http://mathworld.wolfram.com/images/equations/InverseErfc/NumberedEquation4.gif) |
(7)
|
(which follows from the method of Parker 1955).
The Taylor series about 1 is given by
 |
(8)
|
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."
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