Fourier series notes


Fourier series is an expansion of a periodic function of period $2\pi$ which is representation of a function in a series of sine or cosine such as
$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)+\sum_{n=1}^{\infty }b_{n}sin(nx)$

where $a_{0} $, $a_{n}$ and $b_{n}$ are constants and are known as Fourier coefficients.
In applying Fourier theorem for analysis of a complex periodic function, given function must satisfy the following condition
(i) It should be single valued
(ii) It should be continuous.

Drichlet’s Conditions (sufficient but not necessary)

When a function $f(x)$ is to be expanded in the interval (a,b)
(a) $f(a)$ is continuous in interval (a,b) except for finite number of finite discontinuities.
(b) $f(x)$ has a finite number of maxima and minima in this interval.

Orthogonal property of sine and cosine functions

$\int_{-\pi}^{\pi}sin(mx)cos(mx)dx=0$
$\int_{-\pi}^{\pi}sin(mx)sin(nx)dx= \int_{-\pi}^{\pi}sin(mx)sin(nx)dx=\begin{bmatrix}
\pi\delta_{mn} &m\neq 0 \\
0& m=0
\end{bmatrix} $
$\int_{-\pi}^{\pi}cos(mx)cos(nx)dx=\begin{bmatrix}
\pi\delta_{mn} &m\neq 0 \\
2\pi& m=0
\end{bmatrix} $

Fourier Constants

$a_{0}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx$
$a_{0}$ is the average value of function $f(x)$ over the interval
$a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx$
$b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx$

For even functions

$f(-x)=f(x)$ and fourier series becomes
$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)$

For odd functions



$f(-x)=-f(x)$ and fourier series becomes
$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}sin(nx)$
Complex form of fourier series

putting $c_{0}=c_{0}$
$c_{n}=\frac{a_{n}-ib_{n}}{2}$
and
$c_{-n}=\frac{a_{n}+ib_{n}}{2}$
$f(x)=\sum_{-\infty }^{\infty }C_{n}e^{inx}$
coefficent
$C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$

Fourier series in interval (0,T)

General fourier series of a periodic piecewise continous function $f(T)$ having period $T=\frac{2\pi}{\omega}$ is
$f(t)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)+\sum_{n=1}^{\infty }b_{n}sin(nx)$
where
$a_{0}=\frac{1}{T}\int_{0}^{T}f(t)dt$
$a_{n}=\frac{2}{T}\int_{0}^{T}f(t)cos(n\omega T)dt$
$b_{n}=\frac{2}{T}\int_{0}^{T}f(t)sin(n\omega T)dt$
Complex Form of Fourier Series
$f(x)=\sum_{n=-\infty }^{\infty }C_{n}e^{-i\omega t}$
where
$c_{n}=\frac{1}{T}\int_{0}^{T}f(t)e^{-i\omega t}dx$

Advantages of Fourier series

1. It can also represent discontinuous functions
2.  Even and odd functions are conveniently represented as cosine and sine series.
3.  Fourier expansion gives no assurance of its validity outside the interval.

Change of interval from $(-\pi,\pi)$ to $(-l,l)$

Series will be
$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(\frac{nx\pi}{l})+\sum_{n=1}^{\infty }b_{n}sin(\frac{nx\pi}{l})$
with
$a_{0}=\frac{1}{2l}\int_{-l}^{l}f(x)dx$
$a_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)cos(\frac{n\pi x}{l})dx$
$b_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)sin(\frac{n\pi x}{l})dx$

Fourier Series in interval $(0,l)$

Cosine series when function $f(x)$ is even
$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(\frac{n\pi x}{l})$
$a_{0}=\frac{1}{l}\int_{0}^{l}f(x)dx$
$a_{n}=\frac{2}{l}\int_{0}^{l}f(x)cos(\frac{n\pi x}{l})dx$
Sine series when function $f(x)$ is odd
$f(x)=\sum_{n=1}^{\infty }a_{n}sin(\frac{n\pi x}{l})$
$b_{n}=\frac{2}{l}\int_{0}^{l}f(x)sin(\frac{n\pi x}{l})dx$