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)
On the growth of convergence radii for the eigenvalues of the Mathieu equation. (English) Zbl 0912.34027
The author reconsiders the classical Mathieu equation $$y''+ (\mu+ 2\lambda\cos 2x) y=0$$ in the restricted case where it has nontrivial $\pi$-periodic or $\pi$-antiperiodic solutions. In the first case it holds $y(x+\pi)= y(x)$ while in the second case $y(x+ \pi)= -y(x)$, yielding $\mu$ as a function of $\lambda$, say $\mu= \mu_0(\lambda)$ $(n= 1,2,\dots)$ (eigenvalues). The aim is to find a new, sharper estimation for the radius of convergence of the power series of $\mu_n(\lambda)$ about the point $\lambda= 0$. The result, which seems to be the best one, is as follows: $$\lim_{n\to\infty}\inf {\rho_n\over n^2}\ge k k' K^2= 2.041834\dots\ .$$ Here, $\rho_n$ refers the radius of convergence of power series of $\mu_0(\lambda)$ while $K= K(k)$ denotes the complete elliptic integral of the first kind and $k'=(1- k^2)^{1/2}$. As to the modulus $k$, it is determined through the relation $2E= K$, $E$ being the corresponding complete elliptic integral of the second kind.

34B30Special ODE (Mathieu, Hill, Bessel, etc.)
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
33E10Lamé, Mathieu, and spheroidal wave functions
Full Text: DOI