The Titchmarsh integral transformation by the index of a Bessel function

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, Lp-properties of the Titchmarsh transform are investigated and inversion theorems are proved. The question of the correctness of Titchmarshʹs formulas is completely closed by this discussion.