Definition:Laplace Transform/Graphical Interpretation

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Definition

Define $\gamma$ as the integrand of $\displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ 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.


Examples

The following are three examples of the contours defined by the integrand defining the Laplace transform of $\map \cos t$.


Laplace Transform of $\cos t$ of $3 - 9 i$

For $\map {\laptrans {\map \cos t} } {3 - 9 i}$, notice how the contour spirals towards the origin.

Intuitively, the integral converges because as $t$ increases without bound, the contour "shrinks" as it continues its path into the origin.


LCost 3minus9i.png


Laplace Transform of $\cos t$ of $\dfrac 1 2 + 4 i$

For $\map {\laptrans {\map \cos t} } {\dfrac 1 2 + 4 i}$, though the contour is not simple, it still "shrinks" as the parameter $t$ increases without bound:


LCost halfPlus4i.png


Laplace Transform of $\cos t$ of $-3 + 12 i$

From Laplace Transform of Cosine, $\map {\laptrans {\map \cos t} } {-3 + 12 i}$ does not exist.

Notice how the contour spirals outward as $t$ increases without bound, never settling at a "stable point":


LCost minus3plus12i.png


Sources