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Asymptotic inversion of incomplete gamma functions. (English) Zbl 0759.33001
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.

33B15Gamma, beta and polygamma functions
33B20Incomplete beta and gamma functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
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