Real Inversion Formulas for Laplace and Stieltjes Transforms (original) (raw)

On the inversion of the Laplace transform (In Memory of Dimitris Gatzouras)

Cornell University - arXiv, 2022

The Laplace transform is a useful and powerful analytic tool with applications to several areas of applied mathematics, including differential equations, probability and statistics. Similarly to the inversion of the Fourier transform, inversion formulae for the Laplace transform are of central importance; such formulae are old and wellknown (Fourier-Mellin or Bromwich integral, Post-Widder inversion). The present work is motivated from an elementary statistical problem, namely, the unbiased estimation of a parametric function of the scale in the basic model of a random sample from exponential distribution. The form of the uniformly minimum variance unbiased estimator of a parametric function h(λ), as well as its variance, are obtained as series in Laguerre polynomials and the corresponding Fourier coefficients, and a particular application of this result yields a novel inversion formula for the Laplace transform.

Numerical accuracy of real inversion formulas for the Laplace transform

Journal of Computational and Applied Mathematics, 2010

In this article, we investigate and compare a number of real inversion formulas for the Laplace transform. The focus is on the accuracy and applicability of the formulas for numerical inversion. In this contribution, we study the performance of the formulas for measures concentrated on a positive half-line to continue with measures on an arbitrary half-line. As our trial measure concentrated on a positive half-line, we take the broad Gamma probability distribution family.

Complex inversion formula for the distributional Stieltjes transform

Proceedings of the Edinburgh Mathematical Society, 1987

There are several approaches to the Stieltjes transform of generalized functions ([1, 10, 5, 6, 3, 2]). In this paper we use the definition of the distributional Stieltjes transform of index ρ (ρ ∈ ℝ\(−ℕ0); ℕ0 = ℕ∪{0}), Sρ-transform, given by Lavoine and Misra [3]. The Sρ-transform is defined for a subspace of the Schwartz space (ℝ) while in [10, 5, 6, 2] the Stieltjes transform is defined for the elements of appropriate spaces of generalized functions. In these spaces differentiation is not defined which means that the Stieltjes transform of some important distributions, for example δ(k)(x − a), a≧0, k ∈ ℕ, is meaningless in the sense of [10, 5, 6, 2]. It is easy to see that the distributions δ(k)(x − a), a≧0, k ∈ ℕ, have the Sρ-transform for ρ>−k, ρ∈ℝ\(−ℕ0). These facts favour the approach to the Stieltjes transform given in [3].

Recent developments on the Stieltjes transform of generalized functions

International Journal of Mathematics and Mathematical Sciences, 1987

This paper is concerned with recent developments on the Stieltjes transform of generalized functions. Sections and 2 give a very brief introduction to the subject and the Stieltjes transform of ordinary functions with an emphasis to the inversion theorems. The Stieltjes transform of generalized functions is described in section 3 with a special attention to the inversion theorems of this transform. Sections 4 and 5 deal with the adjoint and kernel methods used for the development of the Stieltjes transform of generalized functions. The real and complex inversion theorems are discussed in sections 6 and 7. The Poisson transform of generalized functions, the iteration of the Laplace transform and the iterated Stieltjes transfrom are included in sections 8, 9 and I0. The Stieltjes transforms of different orders and the fractional order integration and further generalizations of the Stieltjes transform are discussed in sections ii and 12. Sections 13, 14 and 15 are devoted to Abelian theorems, initial-value and final-value results. Some applications of the Stieltjes transforms are discussed in section 16. The final section deals with some open questions and unsolved problems. Many important and recent references are listed at the end.

Some new convolution properties and inversion formulas of Laplace transforms

Integral Transforms and Special Functions, 2014

This paper deals with new convolution properties and some new inversion formulas for the Laplace transforms which are not available in the extensive literature of Laplace transforms. The derivation of these inverse Laplace transforms involves new integral formulas of exponential form. These integral formulas are provided in the appendix. The inverse Laplace transform formulas are presented in a tabular form so that they can be used to solve many problems in applied mathematics and engineering science.