# Definition:Laplace Transform/Restriction to Reals

## Definition

Although the definition of the Laplace transform has $s$ be a complex variable, sometimes the restriction of $\map {\laptrans f} s$ to wholly real $s$ is sufficient to solve a particular differential equation.

Therefore, elementary textbooks introducing the Laplace transform will often write something like the following:

... where we assume at present that the parameter $s$ is real. Later it will be found useful to consider $s$ complex.
-- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms: Chapter $1$: The Laplace Transform: Definition of the Laplace Transform

A profound understanding of the workings of the Laplace transform requires considering it to be a so-called analytic function of a complex variable, but in most of this book we shall assume that the variable $s$ is real.
-- 2003: Anders Vretblad: Fourier Analysis and its Applications: $\S 3.1$