Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 32

Definition of the Laplace Transform of $\map F t$

 * $32.1$: Definition of Laplace Transform: $\displaystyle \laptrans {\map F t} = \int_0^\infty e^{-s t} \map F t \rd t = \map f s$

In general $f \left({s}\right)$ will exist for $s > a$ where $a$ is some constant $\mathcal L$ 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 $\map f s$.

$\mathcal L^{-1}$ 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:


 * $32.2$: Definition of Inverse Laplace Transform: $\displaystyle \map F t = \frac 1 {2 \pi i} \int_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s = \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s$

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.