4 Fourier series and Fourier transforms

Abstract The Fourier expansion of an arbitrary piecewise-smooth function in the finite interval —π 〈 x 〈 π as an infinite sum of terms (a 0; a n cos nx, b n sin nx, integer n = l,…,∞) is described. The complex exponent functions e inx, integer —∞ 〈 n 〈 ∞, (or another set of orthogonal functions,) in the interval (a, b) can also be used. A special limiting case for the interval (—∞,∞) yields the Fourier integral transform. It is used to solve inhomogeneous differential equations with the help of Dirac delta functions and Green's functions. The convergence and completeness of Fourier-series representation are discussed. Fourier transforms in multidimensional space-time give access to Fourier spaces of wave vector and frequency. In these Fourier spaces, Maxwell's equations simplify to simple algebraic equations. The Helmholtz decomposition theorem is derived.