public-notes/Laplace Transforms.md

#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_ne^{(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.