# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Asymptotic inversion of incomplete gamma functions. (English) Zbl 0759.33001

The normalized incomplete gamma functions are defined by

$P\left(a,x\right)=\frac{1}{{\Gamma }\left(a\right)}{\int }_{a}^{x}{t}^{a-1}{e}^{-t}dt,\phantom{\rule{2.em}{0ex}}Q\left(a,x\right)=\frac{1}{{\Gamma }\left(a\right)}{\int }_{x}^{+\infty }{t}^{a-1}{e}^{-t}dt,$

where $a>0$, $x\ge 0$. The author is interested in the $x$-values that solves the following (equivalent) equations: $P\left(a,x\right)=p$, $Q\left(a,x\right)=q$, where $a>0$ is fixed, $p\in \left[0,1\right]$ 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 $P\left(a,x\right)$ and $Q\left(a,x\right)$ 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.

##### MSC:
 33B15 Gamma, beta and polygamma functions 33B20 Incomplete beta and gamma functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
##### Keywords:
asymptotic inversion; incomplete gamma functions