Fourier Series

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 an complex periodic function , given function must satisfy 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 continous in interval (a,b) except for finite number of finite discontinuties.
  (b) $f(x)$ has 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 discontinous 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
  $f(x)=\sum_{n=1}^{\infty }a_{n}sin(\frac{n\pi x}{l})$
  $b_{n}=\frac{2}{l}\int_{0}^{l}f(x)sin(\frac{n\pi x}{l})dx$

Complex Analysis Part 2

Liouville's Theorem 
If a function $f(z)$ is analytic for all finite values of z, and is bounded then it is a constant.
Note:- $e^{z+2\pi i} = e^z$

Taylor's Theorem
If a function $f(z)$ is analytic at all points inside a circle C, with its centre at point a and radius R then at each point z inside C
$f(z)=f(a)+(z-a)f'(a)+\frac{1}{2!}(z-a)^2f''(a)+.......+\frac{1}{n!}(z-a)^nf^n(a)$
Taylor's theorem is applicable when function is analytic at all points inside a circle.

 Laurent Series
If $f(z)$ is analytic on $C_{1}$ and $C_{2}$ and in the annular region R bounded by the two concentric circles $C_{1}$ and $C_{2}$ of radii $r_{1}$and $r_{2}$ ($r_{1} > r_{2}$) with their centre at a then for all z inside R
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.........$
where,
$a_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(n+1)}}$
$b_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(-n+1)}}$

Singular points
If a function $f(z)$ is not analytic at point z=a then z=a is known as a singular point or there is a singularity of $f(z)$ at z=a for example
$f(z)=\frac{1}{z-2}$
z=2 is a singularity of $f(z)$

Pole of order m
If $f(z)$ has singularity at z=a then from laurent series expansion
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.....+\frac{b_{m}}{(z-a)^m}+\frac{b_{m+1}}{(z-a)^{m+1}}$
if
$b_{m+1}=b_{m+2}=0$
then
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.....+\frac{b_{m}}{(z-a)^m}$
and we say that function $f(z)$ is having a pole of order m at z=a. If m=1 then point z=a is a simple pole.

Residue
The constant $b_{1}$ , the coefficent of $(z-z_{0})^{-1}$ , in the Laurent series expansion is called the residue of $f(z)$ at singularity $z=z_{0}$
$b_{1}=Res_{z=z_{0}}f(z) = \frac{1}{2\pi i}\int_{C_{1}}f(z)dz$

Methods of finding residues
1.  Residue at a simple pole
if $f(z)$ has a simple pole at z=a then
$Res f(a) =\lim_{z\rightarrow a}(z-a)f(z) $
2. If $f(z)=\frac{\Phi(z)}{\Psi (z)}$
and $\Psi(a)=0$ then
$Res f(a)=\frac{\Phi(z)}{\Psi^{'} (z)}$
3. Residue at pole of order m
If $f(z)$ is a pole of order m at z=a then
$Res f(a)= \frac{1}{(m-1)!}\left \{ \frac{d^{m-1}}{dz^{m-1}}(z-a)^{m}f(z) \right \}_{z=a}$

 Residue Theorem

If $f(z)$ is analytic in closed contour C excapt at finite number of points (poles) within C, then
$\int_{C}f(z)dz = 2\pi i \textsl { [sum of the residues at poles within C]}$

Download this article as pdf

Complex analysis part 2