• What is Laplace transform?
• What's the difference between that and Fourier transform?
• When to use it?

Given a function $f(t)$ on the positive real line $t>0$, we can define the following function of $p$: $$ F(p) = \int_{t=0}^\infty f(t) e^{-pt} dt. $$ Again, we require the function $f(t)$ to have moderate growth at $t \to \infty$ for the integral to be well-defined.

• $f(t) = 1$, $F(p) = 1/p, $ valid for $Re(p) > 0$
• $f(t) = e^{a t}$, $F(p) = 1/(p-a), $valid for $Re(p-a) > 0$.
• $f(t) = \cos(at)$, $F(p) = (1/2)[1/(p-ia) + 1/(p+ia)] = p/(p^2 + a^2). $ valid for $Re(p)>0$ if $a$ is real.

Suppose we know the Laplace transform of $f(t)$, let's denote $F = LT(f)$ (note we just write the name of the function $f$, not including its input variables t). What can we say about $LT(f')$?

We can do integration by part $$ LT(f') = \int_0^\infty e^{-pt} \frac{df}{dt} dt = \int_{t=0}^{t=\infty} e^{-pt} df = \int_{t=0}^{t=\infty} d[e^{-pt} f] - d[e^{-pt}] f= e^{-pt} f(t)|_0^\infty + \int_0^\infty p e^{-pt} f(t) dt = -f(0) + pF(p). $$