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)
Uniform asymptotic approximations for the Whittaker functions $M\sb {\kappa,i\mu}(z)$ and $W\sb{\kappa,i\mu}(z)$. (English) Zbl 1046.33004
Uniform asymptotic approximations are obtained for the Whittaker’s confluent hypergeometric functions $M_{\kappa,i\mu}(z)$ and $W_{\kappa,i\mu}(z)$, where $\kappa$, $\mu$ and $z$ are real. Three cases are considered, and when taken together, result in approximations which are valid for $\kappa\to\infty$ uniformly or $0\le\mu< \infty$, $0< z<\infty$, and also for $\mu\to\infty$ uniformly for $0\le\kappa<\infty$, $0< z<\infty$. The results are obtained by an application of general asymptotic theories for differential equations either having a coalescing turning point and double pole with complex exponent, or a fixed simple turning point. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either modified Bessel functions or Airy functions. Explicit error bounds are available for all the approximations.

33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
41A30Approximation by other special function classes
Full Text: DOI