# 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 w t + a2 · cos 2w t + ... + an · cos n w t + ..
--- | --- | ---
|  | + b1 · sin w t + b2 · sin 2w t + ... + bn · sin n w t + ..
|  | and the Fourier coefficients ak and bk (with the index k = 0, 1, 2, ...) are determined by
ak = 2/T · | T ó õ 0 | f(t) · cos k w t · dt
bk = 2/T · | T ó õ 0 | f(t) · sin k w t · dt
| This can be written much more elegantly using complex numbers and functions as
f(t) = | +¥ S -¥ | cn · einw t
|  | The coefficients cn are obtained by
cn = | T ó õ 0 | f(t) · e– inw t · dt | = | { | ½(an – ibn) |  | for n > 0
½(a–n + ib–n) |  | for n < 0
| The function f(t) is thus expressed as a sum of sin functions with the harmonic frequencies or simply harmonics n·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) · | eiw t · dw
g(w) = | (1/2p) · | +¥ ó õ –¥ | f(t) · e– iw t · dt
| The simplicity, symmetry and elegance (not to mention their usefulness) of these Fourier integrals is just amazing!

With frame

3.3.2 Dipole Relaxation and Dielectric Function

Frequency Dependence of Orietaion Polarization

© H. Föll (Advanced Materials B, part 1 - script)