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

## TL;DR

This paper derives new inverse Laplace transforms for products of parabolic cylinder functions and error functions, providing novel integral representations and correcting existing transforms in the literature.

## Contribution

It introduces new inverse Laplace transforms for products of special functions, correcting previous errors and deriving new integral formulas for hypergeometric functions.

## Key findings

- New inverse Laplace transforms for parabolic cylinder functions
- Corrected existing inverse Laplace transforms in literature
- Derived new integrals for hypergeometric functions

## Abstract

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 two new definite integrals for the generalized hypergeometric function.