Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 32
Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 32
Published $\text {1968}$
$32 \quad$ Laplace Transforms
Definition of the Laplace Transform of $\map F t$
In general $\map f s$ will exist for $s > a$ where $a$ is some constant $\LL$ is called the Laplace transform operator.
Definition of the Inverse Laplace Transform of $\map f s$
If $\laptrans {\map F t} = \map f s$, then we say that $\map F t = \invlaptrans {\map f s}$ is the inverse Laplace transform of $\map f s$.
$\LL^{-1}$ is called the inverse Laplace transform operator.
Complex Inversion Formula
The inverse Laplace transform of $\map f s$ can be found directly by methods of complex variable theory. The result is:
where $c$ is chosen so that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.
Table of General Properties of Laplace Transforms
In the following, $s$ and $t$ are the independent variables of the real functions $f$ and $F$ respectively.
$f$ denotes results of the Laplace transform on functions denoted with $F$.
$a$ and $b$ are constants.