\begin{definition}[Definition:Laplace Transform/Restriction to Reals]
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.''
::::-- {{BookReference|Theory and Problems of Laplace Transforms|1965|Murray R. Spiegel}}: 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.''
::::-- {{BookReference|Fourier Analysis and its Applications|2003|Anders Vretblad}}: $\S 3.1$
\end{definition}