Fourier transform
/ FOOR-yay /
Imagine a single chord struck on a piano. Your ear hears one sound, but a tuner's app shows it split into a few sharp peaks — the individual notes and their loudness. The Fourier transform is the mathematical version of that app: it takes a signal written as a function of time (or space) and tells you how much of every pure frequency it contains. Where the Fourier series did this for a repeating signal using a discrete set of harmonics, the Fourier transform does it for a one-shot, non-repeating signal, and the answer is a smooth curve over all frequencies rather than a list of spikes.
Precisely, the Fourier transform of a function f(x) is F(k) = integral from minus infinity to infinity of f(x) e^{-i k x} dx, where k is the frequency (angular frequency or wavenumber, depending on convention). Each value F(k) measures how strongly the wave e^{i k x} of frequency k is present in f, by sliding that test wave against f and accumulating the overlap. It arises as a limit of the Fourier series: take a periodic function of period L, write its series, and let L grow without bound; the discrete harmonic frequencies n times (2 pi / L) crowd together into a continuum, the sum becomes an integral, and the Fourier coefficients become the continuous function F(k).
This single idea is the backbone of signal processing, optics, quantum mechanics, and the theory of differential equations. It works because of a package of structural rules — linearity, shifting, scaling, the convolution theorem, and the way it turns differentiation into multiplication by i k — that let you carry a hard problem into the frequency domain, solve it by algebra there, and carry the answer back. One honest caveat: the integral as written converges outright only for functions that decay (are absolutely integrable); for sines, steps, and constants you must extend the definition using distributions, where the answer involves the Dirac delta.
The Fourier transform of the Gaussian f(x) = e^{-a x^2} is again a Gaussian: F(k) = sqrt(pi/a) e^{-k^2/(4 a)}. A narrow bump in x (large a) becomes a wide bump in k, and vice versa.
The Gaussian transforms to itself — and the trade between width in x and width in k is the seed of the uncertainty principle.
Conventions differ: where the 1/(2 pi) factor lives and whether the exponent uses k or 2 pi i k vary by textbook and field, so a formula copied from one source can be off by a constant in another — always check the convention before trusting a tabulated transform.