A uniformly convergent series for Sturm-Liouville eigenvalues

Author:
Davis Cope

Journal:
Quart. Appl. Math. **42** (1984), 373-380

MSC:
Primary 34B25

DOI:
https://doi.org/10.1090/qam/757175

MathSciNet review:
757175

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Abstract: For the regular Sturm-Liouville problem with equation on , there are well-known asymptotic expansions for the eigenvalues and eigenfunctions. We show that these asymptotic expansions can be replaced by convergent series for sufficiently large eigenvalues. Convergence is uniform on the interval and uniform with respect to the eigenvalues, in the sense that a single majorant bounds all series. The basic idea is to replace the asymptotic results, which use an expansion of powers of for integers , by a series in powers of , where is an eigenvalue for the corresponding constant coefficient Sturm-Liouville problem with equation .

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DOI:
https://doi.org/10.1090/qam/757175

Article copyright:
© Copyright 1984
American Mathematical Society