(1) Linearity Property
(2) Shifting Property
(3) Change of scale property
(4) Modulation theorem.
(5) nth derivative of the Fourier Transform
(6) Fourier Transform of the derivatives of a function.

**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)} = eisa 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) nth derivative of the Fourier Transform**

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

Then F{xn f(x)} = (-i)n dn/dsn .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 nth 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) = eiax , 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.

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