### Fourier series

(type of series able to approximate a periodic function)

The Fourier series is a type of mathematical series (sum of an infinite sequence of terms) which can be constructed to match a periodic (repeating) function. They are used within many calculation techniques in the physical sciences. It is a means of describing or specifying the function, termed a Fourier series expansion (potentially an alternative means if you already have another) and by using a sum of just some initial terms of the series, offers a means of approximating the periodic function. It does the same for any (real-number) function (of one variable) over a finite interval, that being equivalent to one cycle of some periodic function. Fourier series terms follow a particular form, incorporating coefficients specific to the specific series which can be calculated from values of the function using the Fourier transform. The inverse Fourier transform does the opposite: constructs the function from the Fourier series coefficients.

There are a number of forms of Fourier series which are equivalent and not difficult to convert between: one form has terms consisting of a sine-based function incorporating, in addition to the period, a specific amplitude and phase. Alternative representations express the term as a sum of a sine and cosine (using two amplitudes rather than including a phase) or using the Euler's formula to express each term as a complex exponential function.

Performing mathematical operations on the Fourier coefficients yields the coefficients of other related functions: for example, doubling the amplitude coefficients produces the coefficients of double the original function. This can save work in some cases, and some useful mathematical operations on functions are naturally carried out on the Fourier series coefficients, and difficult or impossible to carry out otherwise. Some computations consist of converting to Fourier series coefficients, carrying out calculations, and converting back.

The Fourier series is the mathematical representation of physical waves in terms of their harmonics: for example, the harmonics (first harmonic, second harmonic, etc.) that you can hear in the sound of a musical instrument are revealed by applying a Fourier transform to data describing the wave form: the Fourier series is the mathematical model for the fact that sound waves are equivalent to the sum of some sine-shaped waves. This equally applies to other waves such as EMR (in its classical representation), ocean waves, etc. Data is sometimes displayed as the Fourier series coefficients derived from the raw data, which offer some insights into the nature of the data.

More complicated series analogous to the (basic) Fourier series handle functions of two or more dimensions. An equivalent type of series for a function over the surface of a sphere (such as "pressure at sea level all over the surface of the Earth", to give an example) are often expressed in terms of a multipole expansion.

(mathematics)