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

Definition of the Laplace Transform of $F \left({t}\right)$

 * $32.1$: Definition of Laplace Transform: $\displaystyle \mathcal L \left\{ {F \left({t}\right)} \right\} = \int_0^\infty e^{-st} F\left({t}\right) \mathrm d t = f \left({s}\right)$

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 $f \left({s}\right)$
If $\mathcal L \left\{{ F \left({t}\right) }\right\} = f \left({s}\right)$, then we say that $F \left({t}\right) = \mathcal L^{-1} \left\{ {f \left({s}\right)} \right\}$ is the inverse Laplace transform of $f \left({s}\right)$.

$\mathcal L^{-1}$ is called the inverse Laplace transform operator.

Complex Inversion Formula
The inverse Laplace transform of $f \left({s}\right)$ can be found directly by methods of complex variable theory. The result is:


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

where $c$ is chosen so that all the singular points of $f \left({s}\right)$ lie to the left of the line $\operatorname{Re} \left({s}\right) = c$ in the complex $s$ plane.