1 Introduction
2 Integral Transforms
3 Fourier Integral Theorem
4 Fourier Transforms And Its Properties
5 Convolution Theorem And Parseval’s Theorem
6 Fourier Sine And Cosine Transforms

**FOURIER TRANSFORMS**

**1 INTRODUCTION**

**2 INTEGRAL TRANSFORMS**

**3 FOURIER INTEGRAL THEOREM**

**4 FOURIER TRANSFORMS AND ITS PROPERTIES**

**5 CONVOLUTION THEOREM AND PARSEVAL’S
THEOREM**

**6 FOURIER SINE AND COSINE TRANSFORMS**

**1 Introduction**

This unit starts with integral
transforms and presents three well-known integral transforms, namely, Complex
Fourier transform, Fourier sine transform, Fourier cosine transform and their
inverse transforms. The concept of Fourier transforms will be introduced after
deriving the Fourier Integral Theorem. The various properties of these
transforms and many solved examples are provided in this chapter. Moreover, the
applications of Fourier Transforms in partial differential equations are many
and are not included here because it is a wide area and beyond the scope of the
book.

**2 Integral Transforms**

The **integral transform** f(s) of a
function f(x) is defined by

if the integral exists
and is denoted by I{f(x)}. Here, K(s,x) is called the **kernel** of the transform. The kernel is a known function of „s‟ and „x‟. The function f(x) is called the **inverse transform **of f(s). By properly
selecting the kernel in the definition of general integral transform, we get various
integral transforms.

The following are some of the well-known transforms:

(i) **Laplace Transform**

(ii)
**Fourier Transform**

(iii) **Mellin Transform**

(iv) **Hankel Transform**

where J_{n}(sx) is the Bessel function of the first kind and order 'n'.

**3
FOURIER INTEGRAL THEOREM**

If f(x) is defined in the interval (-ℓ,ℓ), and the following conditions

(i) f(x) satisfies the Dirichlet‟s conditions in every interval (-ℓ,ℓ),

which is known as the **Fourier integral** of
f(x).

**Note:**

When f(x) satisfies the
conditions stated above, equation (3) holds good at a point of continuity. But
at a point of discontinuity, the value of the integral is (1*/* 2) [f(x+0) +
f(x-0)] as in the case of Fourier series.

**Fourier sine and cosine Integrals**

The Fourier integral of f(x) is given by

When f(x) is an odd function, f(t) coslt is odd while f(t) sinlt is even. Then the first integral of (4) vanishes
and, we get

which is known as the **Fourier sine integral.**

Similarly, when f(x) is an even
function, (4) takes the form

which is known as the **Fourier cosine integral.**

**Complex form of Fourier Integrals**

The Fourier integral of f(x) is given by

Since cos l(t
–x) is an even function of l,
we have by the property of definite integrals

Similarly, since sin l(t
–x) is an odd function of l,
we have

which is the **complex form of the Fourier
integral.**

**4
Fourier Transforms and its properties**

**Fourier Transform**

We know that the complex form of Fourier
integral is

The function F(s),
defined by (1), is called the **Fourier Transform** of f(x). The function
f(x), as given by (2), is called the **inverse Fourier Transform** of F(s).
The equation (2) is also referred to as the **inversion formula**.

**Properties of Fourier Transforms**

**(1) Linearity Property**

If
F(s) and G(s) are Fourier Transforms of f(x) and g(x) respectively, then

F{a f(x) + bg(x)} = a F(s) + bG(s),

where a and b are constants.

= a F(s) + bG(s) i.e,

F{a f(x) + bg(x)} = a F(s) + bG(s)

**(2) ****Shifting
Property **

(i)
If
F(s) is the complex Fourier Transform of f(x), then

F{f(x-a)} = e^{isa} F(s).

(ii) If F(s) is the complex Fourier
Transform of f(x), then

**(3) Change of scale property**

If F(s) is the complex Fourier transform
of f(x), then

F{f(ax)} =1/a F(s/a), a ¹0.

Put
ax = t, so that dx = dt/a.

**(4) Modulation theorem**.

If F(s) is the complex Fourier transform
of f(x),

Then F{f(x) cosax} = ½{F(s+a) + F(s-a)}.

**(5) n ^{th} derivative of the
Fourier Transform**

If F(s) is the complex Fourier Transform of f(x),

Then F{x^{n} f(x)} = (-i)^{n}
d^{n}/ds^{n} .F(s).

**(6) Fourier Transform of the derivatives
of a function**.

If
F(s) is the complex Fourier Transform of f(x),

Then, F{f-isF(s) if„(x)}f(x)®0as x=® ±¥.

In general, the Fourier transform of the
n^{th} derivative of f(x) is given by

F{f ^{n}(x)} = (-is)^{n}
F(s),

provided-1 ‟the derivatives frst®„n±¥. vanish
as x

**Property (7)**

If F(s) is the complex Fourier Transform
of f(x), then F

**Property (8)**

If F(s) is the complex Fourier transform
of f(x),

**Note**: If F{f(x)} = F(s), then

F{f(-x)} = F(-s).

**Example
1**

Find the F.T of f(x) defined by

f(x) = 0 x<a

= 1 a<x<b

= 0 x>b.

The F.T of f(x) is given by

**Example 2**

Find the F.T of f(x) = x for |x
| =< a

= 0 for
|x | > a.

**Example 3**

Find the F.T of f(x) = e^{iax} , 0 < x < 1

= 0 otherwise

The F.T of f(x) is given by

**Note**:

If the F.T of f(x) is f(s), the function
f(x) is called self-reciprocal. In the above example
e ^{-x 2/ 2 }is self-reciprocal under F.T.

**Example 5**

Find the F.T
of

f(x) = 1 for |x|<1.

= 0 for |x|>1.

**5 Convolution Theorem **.** and Parseval’s Theorem**

The convolution of two functions f(x)
and g(x) is defined as

**Convolution
Theorem for Fourier Transforms.**

The Fourier Transform of the convolution
of f(x) and g(x) is the product of their Fourier Transforms,

i.e,
F{f(x) * g(x)} = F{f(x).F{g(x)}.

**Proof:**

F{f(x) * g(x)} = F{(f*g)x)}

Hence, F{f(x) * g(x)} = F{f(x).F{g(x)}.

**Parseval’s**
**identity for Fourier
Transforms**

If
F(s) is the F.T of f(x), then

**Proof:**

By convolution theorem, we have

F{f(x)
* g(x)} = F(s).G(s).

Therefore, (f*g) (x) = F^{-1}{F(s).G(s)}.

**Example
6**

Find
the F.T of f (x) = 1-|x|for |x|<1.

=
0 for |x|> 1

**Example
7**

Find the F.T of f(x) if

f(x) =

= 1 for |x|<a

=
0 for |x|>a>0.

**6 Fourier sine and cosine transforms:**

**Fourier sine Transform**

We know that the Fourier sine integral
is

The function F_{s}(s),
as defined by (1), is known as the **Fourier sine transform** of f(x). Also
the function f(x), as given by (2),is called the **Inverse Fourier sine
transform** of F_{s}(s) .

**Fourier cosine transform**

Similarly, it follows from the Fourier
cosine integral

The function F_{c}(s),
as defined by (3), is known as the **Fourier cosine transform** of f(x).
Also the function f(x), as given by (4),is called the **Inverse Fourier cosine**
**transform **of F_{c}(s) .

**Properties of Fourier sine and cosine
Transforms**

If F_{s}(s) and
F_{c}(s) are the Fourier sine and cosine transforms of f(x)
respectively, the following properties and identities are true.

**(1) Linearity property**

F_{s}
[a f(x) + b g(x) ] = a F_{s} { f(x) } + b F_{s} { g(x) }.

and F_{c} [a f(x) + b g(x) ] = a F_{c}
{ f(x) } + b F_{c} { g(x) }.

**(2) Change of scale property**

F_{s}
[ f(ax) ] = (1/a) F_{s} [ s/a ].

and F_{c} [ f(ax) ] = (1/a) F_{c}
[ s/a ].

**(3) ****Modulation
Theorem **

i.
F_{s} [ f(x) sinax ] = (1/2) [ F_{c}
(s-a) - F_{c} (s+a)].

ii.
F_{s} [ f(x) cosax ] = (1/2) [ F_{s}
(s+a) + F_{s} (s-a)].

iii. F_{c}[
f(x) cosax ] = (1/2) [ F_{c} (s+a) + F_{c} (s-a) ].

iv. F_{c}[
f(x) sinax ] = (1/2) [ F_{s} (s+a) - F_{s} (s-a) ].

**Proof**

The Fourier sine transform of f(x)sinax
is given by

Similarly, we can prove the results (ii), (iii) & (iv).

Similarly, we can prove the second identity and the
other identities follow by setting g(x) = f(x) in the first identity.

**Property
(5)**

If F_{s}(s) and F_{c}(s) are the
Fourier sine and cosine transforms of f(x) respectively, then

**Example
8**

Find the Fourier sine and cosine
transforms of e^{-ax} and hence deduce the inversion formula.

The Fourier sine transform of f(x) is given by

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