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**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

about y-axis.

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

about the origin.

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

Excerpt out of 5 pages

- Quote paper
- Umana Rafiq (Author), 2012, Fourier Series Analysis And Applications, Munich, GRIN Verlag, https://www.grin.com/document/279395

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