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Zbl 0759.33001
Temme, N.M.
Asymptotic inversion of incomplete gamma functions. (English)
[J] Math. Comput. 58, No.198, 755-764 (1992). ISSN 0025-5718; ISSN 1088-6842/e

The normalized incomplete gamma functions are defined by $$P(a,x)={1\over{\Gamma(a)}} \int\sb a\sp x t\sp{a-1} e\sp{-t}dt, \qquad Q(a,x)={1\over{\Gamma(a)}} \int\sb x\sp{+\infty} t\sp{a-1} e\sp{-t} dt,$$ where $a>0$, $x\geq 0$. The author is interested in the $x$-values that solves the following (equivalent) equations: $P(a,x)=p$, $Q(a,x)=q$, where $a>0$ is fixed, $p\in[0,1]$ and $q=1-p$. This problem is of importance e.g. in probability theory and mathematical statistics. The approximations are obtained by using uniform asymptotic expansions of $P(a,x)$ and $Q(a,x)$ in which an error function is the dominant term. Numerical results are indicated and it is shown that the method can be applied also to certain cumulative distribution functions.
[J.Sándor (Jud.Harghita)]

MSC 2000:: *33B15 Gamma-functions, etc.
33B20 Incomplete beta and gamma functions
41A60 Asymptotic problems in approximation
65C99 Numerical simulation

Keywords: asymptotic inversion; incomplete gamma functions

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