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