zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Asymptotische Entwicklungen der Gaußschen hypergeometrischen Funktion für unbeschränkte Parameter. (Asymptotic expansions of the Gauss hypergeometric function for unbounded parameters). (German) Zbl 0732.33005
Asymptotic expansions of the Gauss hypergeometric function ${}\sb 2F\sb 1(a,b;c;z)$ are derived for large absolute values of the complex parameters a,b,c (c$\ne 0,-1,-2,...)$ and for fixed values of the complex variable z $(\vert \arg (1-z)\vert <\pi)$. Assuming $a\sp 2=o(c)$, $b\sp 2=o(c)$ and that Re$\{$ $a\}$ and Re$\{$ $b\}$ are bounded below or two- sided bounded it is shown that the ${\bbfC}$-plane can be divided in two sectors dependent on the value of z so that $$ (i)\quad F(a,b;c;z)\approx \sum\sp{\infty}\sb{\nu =0}\frac{(a)\sb{\nu}(b)\sb{\nu}}{(c)\sb{\nu}}\frac{z\sp{\nu}}{\nu !}, $$ in the sector including the positive real axis $((a)\sb{\nu}$ Pochhammer symbol) and F(a,b;c;z)$\approx$ $$ (ii)\quad \approx \frac{\pi \Gamma (a+b-c)z\sp{1-c}(1-z)\sp{c-b- a}}{\sin (\pi c)\Gamma (1-c)\Gamma (a)\Gamma (b)}\sum\sp{\infty}\sb{\nu =0}\frac{(1-a)\sb{\nu}(1-b)\sb{\nu}(1-z)\sp{\nu}}{(c-b-a+1)\sb{\nu}\nu !} $$ in the remaining sector. In particular, it follows that (i) is not valid for all z with $\vert z\vert <1$, when the complex parameter c tends arbitrary to infinity. This refutes an assertion in the well-known book “Higher transcendental functions”, Vol. 1 by {\it A. Erdélyi} et. al. (1951; Zbl 0051.303).
Reviewer: E.Wagner

33C20Generalized hypergeometric series, ${}_pF_q$
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)