Definition:Laplace Transform/Graphical Interpretation

Definition
Define $\gamma$ as the integrand of $\ds \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ as a function of $\map f t$, $s$, and $t$:


 * $\map \gamma {\map f t, t; s} = \map f t \, e^{-s t}$

For any particular function $f$, holding $s$ fixed, the integrand of the Laplace Transform $\kappa$ can be interpreted as a contour.

That is, for a given function $f$ and a particular complex number $s_0$ held constant:


 * $\map \kappa t: \R_{\ge 0} \to \C$


 * $\map \kappa t = \map \gamma {\map f t, t; s_0}$

is a parameterization of a contour.