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
- 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: $\S 3.1$
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: $\S 3.1$
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- 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.): $\S 6.1$