Definition:Inverse Laplace Transform/Definition 2

Definition
Let $f \left({s}\right): S \to \R$ be a complex function, where $S \subset \R$.

The inverse Laplace transform of $f$, denoted $F \left({t}\right): \R \to S$, is defined as:


 * $\displaystyle F \left({t}\right) = \frac 1 {2 \pi i} \operatorname{PV}\int_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} f \left({s}\right) \ \mathrm d s = \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} f \left({s}\right) \ \mathrm d s$

where $\operatorname{PV}\displaystyle \int$ is the Cauchy principal value of the integral.

Here $c$ is any real constant such 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.