# A generalization of Laplace and Fourier transforms

## TL;DR

This paper introduces the symmetric Laplace transform, a new mathematical tool that unifies Laplace and Fourier transforms, with applications to differential equations and parabolic problems.

## Contribution

It defines the symmetric Laplace transform, explores its properties, and demonstrates its usefulness in solving differential equations, extending the capabilities of existing transforms.

## Key findings

- The symmetric Laplace transform combines features of Laplace and Fourier transforms.
- Basic properties and inverse form are established.
- Applied to solve a parabolic problem and an ODE.

## Abstract

In this note we propose a generalization of the Laplace and Fourier transforms which we call symmetric Laplace transform. It combines both the advantages of the Fourier and Laplace transforms. We give the definition of this generalization, some examples and basic properties. We also give the form of its inverse by using the theory of the Fourier transform. Finally, we apply the symmetric Laplace transform to a parabolic problem and to an ordinary differential equation.