Application of the Efros theorem to the function represented by the inverse Laplace transform of $s^{-μ}\,\exp(-s^ν)$

Some Laplace transforms and integral representations for parabolic cylinder functions and error functions

This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently specialized for products of the error function and its complement thereby yielding new integral representations for products of the latter two functions. The transforms that are derived in this paper also allow to correct two inverse Laplace transforms that are widely reported in the literature and subsequently uses one of the corrected expressions to obtain …

Higher Asymptotics of Laplace's Approximation

We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the integrand. Our most general result assumes no smoothness of the functions of the integrand, but the expressions we obtain contain integrals which may be difficult to evaluate in practice. We then make additional assumptions which are sufficient to simplify these integrals, in some cases obtaining explicit formulae for the coefficients in the asymptotic expansions…

The Laplace transform of the digamma function: an integral due to Glasser, Manna and Oloa

We provide an analytic expression for the Laplace transform of the digamma fuction. This complements work of L. Glasser, D. Manna and O. Oloa on this question. The Laplace transform is continuous in the Laplace-variable a. The derivative admits a jump at a= ln 2.

A Tauberian Theorem for Laplace Transforms with Pseudofunction Boundary Behavior

The prime number theorem provided the chief impulse for complex Tauberian theory, in which the boundary behavior of a transform in the complex plane plays a crucial role. We consider Laplace transforms of bounded functions. Our Tauberian theorem does not allow first-order poles on the imaginary axis, but any milder singularities, characterized by pseudofunction boundary behavior, are permissible. In this context we obtain a useful Tauberian theorem by exploiting Newman's `contour method'.

Product of parabolic cylinder functions involving Laplace transforms of confluent hypergeometric functions

In this paper, the product of parabolic cylinder functions $D_ν(\pm z)D_{ν+μ-1}(z)$, with different parameters $μ$ and $ν$, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeometric functions. The provided integral representations are transformed to easily yield Nicholson-type integral forms and used to derive other series expansions for products of parabolic cylinder functions.

Laplace积分的高阶渐近展开及其误差估计

Higher-order asymptotic expansions for Laplace's integral and their error estimates

本文研究Laplace积分的渐近分析，推导了带有误差估计的高阶渐近展开公式以推广前人结果，并比较了这些公式与数值积分的近似效果。