# 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)
Application of uniform asymptotics to the second Painlevé transcendent. (English) Zbl 0912.34007
Connection problems of Painlevé differential equations of the form ${d}^{2}\phi /d{\eta }^{2}=-{\xi }^{2}F\left(\eta ,\xi \right)\phi$ are studied. These problems involve finding uniform approximations to solutions to this equation when the independent variable passes towards infinity along different directions in the complex plane. By the method used the need to match solutions is avoided. The treatment depends on the locations of the zeros of the function $F$ in the limit. If they are isolated a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. If two of the zeros of $F$ coalesce as $|\xi |\to \infty$ then an approximation can be derived in terms of parabolic cylinder functions.
Reviewer: V.Burjan (Praha)

##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 34A45 Theoretical approximation of solutions of ODE 34M55 Painlevé and other special equations; classification, hierarchies 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34M40 Stokes phenomena and connection problems (ODE in the complex domain)