Fourier Series

Fourier series are made up of sinusoids, all of which have frequencies that are integer multiples of some fundamental frequency. The trick, as with Taylor series, is to figure out what the coefficients are. In summation notation, we say (for odd functions of period 2, but that's just being picky in this context):
. . . and the trick is finding the coefficients ak. You can find those coefficients by using calculus on complex exponentials, or you can use NuCalc and just build your function out of sines.
A great thing about using Fourier series on periodic functions is that the first few terms often are a pretty good approximation to the whole function, not just the region around a special point. Fourier series are used extensively in engineering, especially for processing images and other signals. Finding the coefficients of a Fourier series is the same as doing a spectral analysis of a function.
In mathematics, a Fourier series decomposes a periodic function of periodic signal into a sum of simple oscillating functions, namely sine and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.
The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combiantion) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.
Although the original motivation was to solve the heat equation it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The basic results are very easy to understand using the modern theory