The Titchmarsh integral transformation by the index of a Bessel function

• Type: Article in International Scientific Journal
• Date: 2000

Authors

Yakubovich, SB (Author) FCUP

Journal

Title: Journal of Computational and Applied Mathematics

Vol. 118

Pages: 353-361

ISSN: 0377-0427

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 4 Citations

Scopus - 4 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-000-ZXH

DOI: 10.1016/s0377-0427(00)00298-3

Abstract (EN): This paper deals with an index integral transformation using Bessel functions as kernels. It was introduced and studied by Titchmarsh in 1946 as an example of a continuous spectrum Bessel-function expansions in Sturm-Liouville boundary value problems. Later in the second edition of his book (Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations, Part I, 2nd Edition, Clarendon Press, Oxford, 1946) in 1962 he corrected his expansion by adding an additional term, which contains a combination of an integral and series. In this paper the Titchmarsh formula is simplified and contains just integrals with Bessel and Lommel functions as kernels, which generate a pair of Titchmarsh integral transformations. By using the composition properties of the Titchmarsh transform and its relationship with the Kontorovich-Lebedev transform, L-p-properties of the Titchmarsh transform are investigated and inversion theorems are proved. The question of the correctness of Titchmarsh's formulas is completely closed by this discussion.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 9