Fourier series

In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), refers to an infinite series representation of a periodic function ƒ of a real variable ξ, of period P:

$f(\xi +P)=f(\xi )\ .$

In the case of a complex-valued function ƒ(ξ), Fourier's theorem states that an infinite series, known as a Fourier series, is equivalent (in some sense) to such a function:

$f(\xi )=\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi in\xi /P}$

where the coefficients {c_n} are defined by

$c_{n}={\frac {1}{P}}\int _{0}^{P}f(\xi )\exp \left({\frac {-2\pi in\xi }{P}}\right)\,d\xi \ .$

In what sense it may be said that this series converges to ƒ(ξ) is a complicated question. However, physicists being less delicate than mathematicians in these matters, simply write

$f(\xi )=\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi in\xi /P}\ ,$

and usually do not worry too much about the conditions to be imposed on the arbitrary function ƒ(ξ) of period P in order that this expansion converge to the function.

Real-valued functions in time and space

(PD) Image: John R. Brews
The first six terms of a Fourier series for a periodic sawtooth wave (bottom) and their summation (top). The sum oscillates about the value of the function, a behavior called Gibbs phenomenon

Fourier's theorem states that any real-valued periodic function can be expressed as a sum of sinusoidal functions with periods related to P:

$f(\xi )=a_{0}+\sum _{1}^{\infty }a_{n}\cos \left({\frac {2\pi }{P/n}}\xi +\varphi _{n}\right)\ ,$

a series of cosines with various phases {φ_n}. Using the cosine relation:

$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)\ ,$

and the orthogonality relations:

${\frac {2}{P}}\int _{0}^{P}\ d\xi \cos \left({\frac {2\pi }{P/n}}\xi \right)\cos \left({\frac {2\pi }{P/m}}\xi \right)=\delta _{n,m}\ ,$

${\frac {2}{P}}\int _{0}^{P}\ d\xi \sin \left({\frac {2\pi }{P/n}}\xi \right)\sin \left({\frac {2\pi }{P/m}}\xi \right)=\delta _{n,m}\ ,$

${\frac {2}{P}}\int _{0}^{P}\ d\xi \cos \left({\frac {2\pi }{P/n}}\xi \right)\sin \left({\frac {2\pi }{P/m}}\xi \right)=0\ ,$

one finds:

$a_{0}={\frac {1}{P}}\int _{0}^{P}\ d\xi \ f(\xi )$

$a_{n}\cos \varphi _{n}={\frac {2}{P}}\int _{0}^{P}\ d\xi \ f(\xi )\cos \left({\frac {2\pi }{P/n}}\xi \right)$

$a_{n}\sin \varphi _{n}=-{\frac {2}{P}}\int _{0}^{P}\ d\xi \ f(\xi )\sin \left({\frac {2\pi }{P/n}}\xi \right)\ ,$

thereby determining the coefficients {a_n} and the phases {φ_n}.

Thus, a function periodic in time with period T can be expressed as a Fourier series:

$f(t)=a_{0}+\sum _{1}^{\infty }a_{n}\cos \left(n\omega _{0}t+\varphi _{n}\right)\ ,$

where ω₀ = 2π/T is called the fundamental frequency and its multiples 2ω₀, 3ω₀,... are called harmonic frequencies and the cosine terms are called harmonics of ƒ. A function ƒ(x) of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral sub-multiples of λ (i.e. λ, λ/2, λ/3, etc.):

$f(x)=a_{0}+\sum _{1}^{\infty }a_{n}\cos \left({\frac {2\pi }{\lambda /n}}x+\varphi _{n}\right)\ .$

If the function is a fixed waveform propagating in time, we may take ξ as:

$\xi =x-vt\ ,$

where x is a position in space, v is the speed of propagation and t is the time. The period in space at a fixed instant in time is called the wavelength λ=P, and the period in time at a fixed position in space is called the period T=λ/v.