Laplace Transforms

Calculus Math

Background - Analytic Continuation

$${\int }_0^{\infty }{e}^{-st}dt=\frac{1}{s}$$

is used as an analytic continuation of the function. For the Laplace Transform to work, most of the integrals used must be extended to analytic continuations.

Definition - Laplace Transform

$$F(s)={\int }_0^{\infty }f(x)e^{-st}dt$$

Intuition - The $e^{sx}$ Finding Machine

Take $f(x)$ as $\sum c_n{e}^{at}$. Plugging into the Laplace Transform:

$$F(s)={\int }_0^{\infty }\sum c_n{e}^{(a-s)t}dt$$ $$=\sum c_n{\int }_0^{\infty }{e}^{-(s-a)t}dt$$ $$=\sum \frac{c_n}{s-a}$$

Therefore the Laplace Transform of a function reveals both $c_n$ and $s$ in the sum based upon the parts that make up the transform: poles reveal all $s$ values, while the “magnitude” of each pole reveals the magnitude of each $e^{sx}$ term.