# Definition:Laplace Transform/Restriction to Reals

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## 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

- -- 1965: Murray R. Spiegel:

*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*: $\S 3.1$

- -- 2003: Anders Vretblad:

## Sources

- 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Definition of the Laplace Transform - 2003: Anders Vretblad:
*Fourier Analysis and its Applications*: $\S 3.1$

- 2004: James Ward Brown and Ruel V. Churchill:
*Complex Variables and Applications*(7th ed.): $\S 7.81$ - 2009: William E. Boyce and Richard C. DiPrima:
*Elementary Differential Equations and Boundary Value Problems*(9th ed.): $\S 6.1$