Fourier Series Analysis And Applications

Scientific Essay, 2012

Excerpt

1
Fourier Series Analysis And Applications
Umana Rafiq Ananna
Department of EEE, Ahsanullah University of Science and Technology, Dhaka, Bangladesh.
Abstract : Fourier Series, Fourier Analysis, Euler's Formula for
Coefficients, Periodic Functions, Trigonometric Series, Even
Function, Odd Function, Properties of Functions, Fourier's
Cosine And Sine Series, Half Range Fourier Sine and Cosine
Series, Examples, Complex form, Riemann-Zeta Function,
Mathematical analysis, Perseval's Formula, Piecewise smooth
function, Bessel's inequality, Riemann lemma, Perseval's
Theorem, Propositions and Remarks, Gibbs Phenomenon,
Physical Applications, Heat distribution in a metal plate, Square
wave, Sawtooth wave, Full an Half wave Rectifier, Advantages
and Conclusion.
1. Introduction
Fourier Series is the founding principle behind the field of
Fourier Analysis, is an infinite expansion of a function in
terms of sines and cosines. In Physics and engineering,
expanding functions in terms of sines and cosines are useful
because it allows one to more easily manipulate functions that
are, for example, discontinuous or simply difficult to represent
analytically. It decomposes periodic functions or periodic
signals into the sum of a (possibly infinite) set of simple
oscillating functions.
The Fourier series is named in honour of Jean-Baptiste Joseph
Fourier (1768­1830), who made important contributions to
the study of trigonometric series, after preliminary
investigations by Leonhard Euler, Jean le Rond d'Alembert,
and Daniel Bernoulli. Fourier introduced the series for the
purpose of solving the heat equation in a metal plate.
These series are very powerful tools in connection with
various problems involving Ordinary and Partial Differential
equations. And have many more unique and undeniable real
life applications which will be discussed as we proceed
further.
2. Definitions
2.1 Periodic Functions
Let f(x) be a real valued function said to be periodic if these
exist in a non-zero number t, independent of x, such that the
equation f(x+t) = f(x) holds for all values of x. The least
value of t0 is called the least period or simply the period
f(x).
Example: Let f(x)=sinx. Then f(x) is a periodic function
having period 2.
Proof:
Here, f(x) = sinx
f(x+2) = sin(x+2) = sin(2+x) = sinx
f(x+4) = sin(x+4) = sin(4+x) = sinx
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
f(x+2n) = sin(x+2n) = sin(2n+x) = sinx [nN]
f(x) = f(x+2) = f(x+4) = . . . . . . . . . = f(x+2n) = sinx
Thus f(x) = sinx is a periodic function having period 2 .
Similarly we can show that f(x) = cosx and f(x) = tanx are
periodic functions having period 2 and respectively.
Figure1: The graph of sine and cosine function
2.2 Trigonometric Series
Any series of the term
a cosnx
b sinnx
where a
o
, a
n
and b
n
are coefficients and constants.
2.3 Fourier Series
The Trigonometric series
f x
a
a cosx
a cos2x
a cosnx
b sinx
b sin2x
b sinnx
f x
a
a cosnx
b sinnx
is called a Fourier Series if the coefficients a
o
, a
n
and b
n
are
defined by the following formula which is called Euler's
Formula for Coefficients ,
a
f x
­
a
f x cosnxdx
­
b
f x sinnxdx
­
where, f(x) is any single valued function defined on the
integral (- , ). The Fourier series can also be written as
a cosnx
b sinnx
where,
a
f x
­
, a
f x cosnxdx
­
,
b
f x sinnxdx
­

2
2.4 Even Function
A function f(x) is said to be even if f(-x)=f(x) is symmetrical
Example:
Let, f(x)=x
2
f(x)=(-x)
2
= x
2
f(x) is an even function.
Properties Of Even Function: If f(x) is Even then,
i.
f x dx
­
2
f x dx
ii.
a
f x dx
iii.
a
f x cosnxdx
iv.
b
0
2.5 Odd Function
A function f(x) is said to be even if f(-x)=f(x) is symmetrical
Let, f(x)= x
f(-x)=-f(x)=-x So, f(x) is an odd function.
Properties Of Odd Function: If f(x) is an odd function then,
i.
a
0
ii.
a
0
iii.
b
f x sinnxdx
2.6 Half Range Fourier's Cosine And Sine Series
When Fourier Series only has the Cosine terms or only the
Sine terms we call such series as Half Range Fourier Cosine
Series or Half Range Sine Series respectively. And such
function must be defined in the integral (0,), which is half of
the (- , ) and the function is specified as odd or even so that
it is clearly defined in the other half of the interval ,namely, (-
,0). In such a case we have,
a
f x cosnxdx for half range cosine series and
b
f x sinnxdx for half range sine series
2.7 Dirichlet's Condition For Fourier Series
If a function f(x) for the interval (- , )
i. is single valued
ii. is bounded
iii. has at most a finite number of maxima and minima
iv. Has only a finite number of discontinuous
v. is f(x)+2 =f(x) for values of x outside (- , ) then,
Sp x
a cosnx
b sinnx converges to f(x) as
p at values of x for which f(x) is continuous and to
f x
0
f x
0 at points of discontinuity.
2.8 Perseval's Formula
f x
dx = c{
a
a
b }
This is known as Perseval's formula.
1. If 0 x 2c, then
f x
dx
c
a
n
b
n
2. F 0 x c then (Half range cosine series) :
f x
dx
c
a
3. F 0 x c then (Half range sine series) :
f x
dx
c
b
4. R.M.S =
3. Work Examples
3.1 Fourier Series in Complex form
We Know,
cos(x) = [e
ix
+ e
-ix
]
sin(x) = [e
ix
- e
-ix
]
Now Fourier Series of a function f(x) of period 2l is,
f x
a
0
2
a
n
cos
nx
l
b
n
sin
nx
l
Now putting the value of cosx and sinx we get,
f x
a
0
2
a
n
1
2
e
e
b
n
1
2i
e
e
Solving this equation we can get,
C
n
1
2
a
n
ib
n
C
n
1
2
a
n
ib
n
where, C
0
a
0
2
1
2
1
l
f x dx
2l
0
C
n
1
2
1
l
f x
2l
0
cos
nx
l
dx
i
l
f x
2l
0
sin
nx
l
dx
Hence,
C
n
1
2l
f x e
inx
l
dx
2l
0
and C
n
1
2l
f x e
inx
l
dx
2l
0
3.2 Riemann Zeta Function
Let f(x) = x
2
in the interval [-,] .This is an even function so
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Details

Title
Fourier Series Analysis And Applications
A
Author
Year
2012
Pages
5
Catalog Number
V279395
ISBN (eBook)
9783656731290
ISBN (Book)
9783656731283
File size
815 KB
Language
English
Tags
fourier, series, analysis, applications
Quote paper
Umana Rafiq (Author), 2012, Fourier Series Analysis And Applications, Munich, GRIN Verlag, https://www.grin.com/document/279395 