Laplace Transform/Graphical Interpretation/Examples

Examples of Graphical Interpretation of Laplace Transforms

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.

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:

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":