Fouriersarja

Fouriersarja, or Fourier series, is a method for expressing a periodic function as an infinite sum of trigonometric terms (sines and cosines) or, equivalently, as a sum of complex exponentials. The idea is to decompose a signal into its harmonics, with frequencies that are integer multiples of a fundamental frequency.

Historically, the theory was developed by Jean-Baptiste Joseph Fourier in the early 19th century to solve problems in heat conduction. The Fourier framework has since become fundamental in mathematics, physics, and engineering for analyzing periodic phenomena.

For a real-valued function f with period 2π that is integrable, its Fourier series is f(x) ~ a0/2 + sum_{n=1}^∞ [a_n cos(nx) + b_n sin(nx)], where a_n = (1/π) ∫_{-π}^{π} f(x) cos(nx) dx and b_n = (1/π) ∫_{-π}^{π} f(x) sin(nx) dx. More generally, for period T, with ω0 = 2π/T, one writes f(x) ~ sum_{n=-∞}^{∞} c_n e^{inω0 x}, where c_n = (1/T) ∫_{0}^{T} f(x) e^{-inω0 x} dx.

Convergence depends on conditions on f. Under Dirichlet conditions, the series converges to the value between left and right limits at a point of discontinuity and to f(x) at points of continuity. In L^2, the series converges to f in mean square. Partial sums may exhibit Gibbs phenomenon near jumps.

Applications include analyzing periodic signals, solving partial differential equations such as the heat and wave equations, and in data processing and image compression by representing functions with a finite number of harmonics. Variants include using cosine or sine series for even or odd functions, or the complex form for theoretical work.