# Fourier Series and Transforms

Fourier Series

Every (physically sensible) periodic function f(t) = f(t + T) with T = 1/n = 2p/w and n, w = frequency and angular frequency, respectively, may be written as a Fourier series as follows

f(t)  »   a0/2 + a1 · cos wt + a2 · cos 2wt + ... + an · cos nwt + ..
+ b1 · sin wt + b2 · sin 2wt + ... + bn · sin nwt + ..

and the Fourier coefficients ak and bk (with the index k = 0, 1, 2, ...) are determined by

ak  =  2/T · T ó õ 0

bk  =  2/T · T ó õ 0

This can be written much more elegantly using complex numbers and functions as

f(t)  = +¥ S -¥

The coefficients cn are obtained by

cn  =  f(t) · e– inwt   · dt T ó õ 0 {

The function f(t) is thus expressed as a sum of sin functions with the harmonic frequencies or simply harmonics w derived from the fundamental frequency w0 = 2p/T.
The coefficients cn define the spectrum of the periodic function by giving the amplitudes of the harmonics that the function contains.

Fourier Transforms

A nonperiodic function f(t) ("well-behaved"; we are not looking at some abominable functions only mathematicians can think of) can also be written as a Fourier series, but now the Fourier coefficients have some values for all frequencies w, not just for some harmonic frequencies.
Instead of a spectrum with defined lines at the harmonic frequencies, we now obtain a spectral density function g(w), defined by the following equations

f(t)  = +¥ ó õ –¥

g(w)  = +¥ ó õ –¥

The simplicity, symmetry and elegance (not to mention their usefulness) of these Fourier integrals is just amazing!

To index

3.3.2 Dipole Relaxation and Dielectric Function

Frequency Dependence of Orietaion Polarization