Laplace transform
/ luh-PLAHSS /
Imagine you have a complicated machine described by a differential equation — a circuit, a vibrating beam, a thermostat — and the calculus of it (derivatives, integrals) is hard to push through by hand. The Laplace transform is a change of viewpoint that trades that calculus for ordinary algebra. You feed in a function of time, f(t), and out comes a function of a new variable s; in that new picture, differentiating becomes multiplying, so a differential equation collapses into something you can solve by rearranging symbols.
Concretely, the Laplace transform of f(t) is defined by the integral L{f}(s) = integral from 0 to infinity of e^{-s t} f(t) dt, usually written F(s). The factor e^{-s t} is a weight that decays in time, and integrating against it squeezes the entire history of f from t = 0 onward into a single number for each value of s. For example, the transform of the constant 1 is 1/s, the transform of e^{a t} is 1/(s - a), and the transform of sin(omega t) is omega/(s^2 + omega^2). Because the lower limit is 0, this is the one-sided transform, which is exactly what you want for problems that start at a definite instant.
This is the workhorse of engineering mathematics. It is the standard route to solving linear constant-coefficient initial-value problems, analyzing circuits in the s-domain, and defining the transfer function of a system. Its power comes from the structural rules it obeys (linearity, the shifting theorems, and especially the transforms of derivatives and integrals), which let you transform a whole problem rather than just look terms up one at a time.
L{e^{a t}}(s) = integral from 0 to infinity of e^{-s t} e^{a t} dt = integral from 0 to infinity of e^{-(s - a) t} dt = 1/(s - a), valid for s > a.
A direct integration shows the exponential becomes a simple pole at s = a — the single most useful pair in the table.
The integral only converges for s in a half-plane (its region of convergence); outside that strip F(s) is defined by analytic continuation, and forgetting the convergence condition is the most common beginner slip.