the Fourier transform
/ FOOR-yay /
Any signal in time, or any shape in space, can be rebuilt as a sum of pure sine waves of different frequencies. The Fourier transform is the prism: it takes a function and tells you exactly how much of each frequency it contains. A sharp spike contains all frequencies; a pure tone contains one. Physics lives in this dual -- sometimes position or time is the natural language, sometimes momentum or frequency.
The Fourier transform of f(x) is F(k) = the integral from -infinity to infinity of f(x) e^(-i k x) dx, and the inverse rebuilds it as f(x) = (1/(2 pi)) times the integral of F(k) e^(i k x) dk (conventions differ on where the 2 pi sits). Its power is that it diagonalizes differentiation: the transform of df/dx is (i k) F(k), so a differential operator becomes multiplication by a polynomial in k. Convolution in one domain becomes ordinary multiplication in the other (the convolution theorem), and Parseval's theorem says total energy is the same computed in either domain.
It is everywhere: in quantum mechanics the position and momentum wavefunctions are Fourier transforms of one another, which is why the width in x times the width in k has a fixed minimum -- the mathematical root of the Heisenberg uncertainty principle. It defines the spectrum in optics and signal processing, turns constant-coefficient PDEs into algebra, and its discrete cousin (the FFT) is one of the most used algorithms in all of computation.
The Fourier transform of a narrow Gaussian e^(-a x^2) is another Gaussian, proportional to e^(-k^2/(4a)). Make the first narrow (large a) and the second is wide -- localize in x and you spread in k, the transform-pair face of the uncertainty principle.
A narrow Gaussian transforms to a wide one, and vice versa.
A strict Fourier transform needs f to decay (be integrable); a pure sine wave or a constant has none in the ordinary sense -- you recover them only by admitting the Dirac delta as a generalized function.