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)
The Lamé wave equation. (English) Zbl 0715.34008

Separation of the three-dimensional Helmholtz equation (Δ+k 2 )u=0 in elliptic coordinates leads to the Lamé wave equation

(1)[f 1/2 (z)d dzf 1/2 (z)d dz+1 4q(z)]w=0

where f(z)=(z-a 1 )(z-a 2 )(z-a 3 ),q(z)=h-z+k 2 z 2 , h, separation constants. In § 2, the spectrum of equation (1) and the Lamé wave functions are defined and a survey on known results is given. The Klein-Bocher classification of linear second order equations with rational coefficients is presented and the position of equation (1) is indicated. In § 3, the asymptotics of the solutions in the complex z-plane are studied for arbitrary complex parameter values k,h,. In § 4, the asymptotics of the spectrum and of the angle wave functions are studied by passing to the complex domain. For the spectral parameters h, a system of equations F j (h,)=πh j +b j +o(1), j=1,2, is obtained where the h j are integers which are generalizations of the classical Bohr-Sommerfeld quantization law. The functions F j are periods of the hyperelliptic integral S(z)= z 0 z [q(t)/f(t)] 1/2 dt.

34M99Differential equations in the complex domain
34B05Linear boundary value problems for ODE
33E10Lamé, Mathieu, and spheroidal wave functions