Fourier sine & Cosine transforms

FOURIER SINE & COSINE TRANSFORMS :

a. FOURIER COSINE TRANSFORM:

The infinite Fourier cosine transform of f (x) is defined by

The inverse Fourier cosine transform Fc [f(x)] is defined by

b. INVERSION FORMULA FOR FOURIER COSINE TRANSFORM

Let Fc(s) denote the F.C.T of f(x). Then

Proof : By the definition of F.C.T,

Here, f(x) is defined for all x ≥ 0

Clearly,

g(-x) = g(x) for all x and hence g is an even function.

To prove that, the Fourier transform of g(x) is the F.C.T of f(x).

Hence, by inversion formula for F.T, we have

c. FOURIER SINE TRANSFORM:

The infinite Fourier sine transform of f(x) is defined by

The inverse Fourier sine transform of Fs [f(x)] is defined by

d. INVERSION FORMULA FOR FOURIER SINE TRANSFORM 

Let Fs(s) denote the F.S.T of f(x). Then

Proof : By the definition of F.S.T

Here, f(x) is defined for all x ≥ 0

Now, we define g(x) by

Clearly, g(x) is an odd function

F(g(x)] = Fs [f(x)] [by the above 5.8]

e. Properties of Fourier sine transform and Fourier cosine transform

1. Linear property

Proof: (i) We know that,

(ii) We know that,

2. Modulation property:

Proof :

4. Fs [f'(x)] = -s F'c (s), if f (x) → 0 as x→ ∞.

[Transform of derivative]

Proof : 

5.

 [Transform of derivative]

Proof : 

6.

 [Derivatives of transform]

Proof: We know that,

Differentiating both sides w.r.to 's', we get

7.

Proof: We know that,

Differentiating both sides w.r.to 's', we get

Ill(a). Problems based on Fourier Cosine Transform

Example 4.3.a(1) : Find the Fourier cosine transform of

Solution :

Example 4.3.a(2) : Find the Fourier cosine transform of and hence, find

Solution: We know that,

By integrating, we get

Example 4.3.a(3): Find the Fourier cosine transform of e-ax, a > 0.

Solution :

Example 4.3.a(4) : Find the Fourier cosine transform of the function 3e-5x + 5e-2x.

Solution : 

Example 4.3.a(5): Find the Fourier cosine transform of

Solution: We know that,

Example 4.3.a(6): Find the Fourier cosine transform of f(x) = x.

Solution : We know that,

Example 4.3.a(7): Find the Fourier cosine transform of e-ax cos ax.

Solution: We know that,

Example 4.3.a(8): Show that is self-reciprocal under Fourier cosine transform.

Solution :

We know that,

Hence,  is self reciprocal with respect to Fourier cosine transform.

Example 4.3.a(9): Find the Fourier cosine transform of e-ax sin ax.

Solution: We know that,

Example 4.3.a(10): Evaluate Fc [xn-1] if 0 < x < 1.

Deduce that is self reciprocal under Fourier cosine transform.

Solution : 

Equating real parts, we get

Using this in (1) we get

Example 4.3.a(11): Find the Fourier cosine transform of

Solution: We know that,

Example 4.3.a(12): Find the Fourier cosine transform of

Solution :

We know that,

Example 4.3.a(13): Find the Fourier cosine transform of

Solution: We know that,

Ill. (b) Problems based on Fourier cosine transform and its inversion formula.

Example 4.3.b(1): Solve the integral equation

Solution: 

Example 4.3.b(2): Solve the integral equation

Solution :

We know that, Fourier cosine inversion formula is

We know that,

From (1) and (2), we have

Example 4.3.b(3): Find the Fourier cosine transform of e-|x| and deduce that

Solution: 

Now, using Fourier cosine inversion formula

Example 4.3.b(4): Find the Fourier cosine transform of e-ax, a > 0 and deduce that

Solution: 

Applying the inversion formula, we have

III. (c) Problems Based on Fourier Sine Transform. [F.S.T]

Example 4.3.c(1): Find the Fourier sine transform of

f(x) = e-x cos x.

Solution: We know that,

Example 4.3.c(2): Find the Fourier sine transform of

Solution: We know that,

Example 4.3.c(3): Find the Fourier sine transform of

Solution: We know that,

Example 4.3.c(4): Find the Fourier sine transform of 1/x.

Solution: We know that,

Example 4.3.c(5): Find the Fourier sine transform of 3e-5x + 5e-2x

Solution: 

Example 4.3.c(6): Find the Fourier sine transforms of f(x) = e-ax 

Solution: 

Example 4.3.c(7): Find the Fourier sine transform of the function

Solution: We know that,

Diff. w.r.to s on both sides,

Example 4.3.c(8): Find the Fourier sine transform of xn-1. Deduce that is self reciprocal under Fourier sine transform.

Solution: 

Example 4.3.c(9): Find the Fourier sine transform of

Solution: 

III. (d) Problems based on Fourier sine transform and its inversion formula.

Example 4.3.d(1): Find Fourier sine transform of e-ax, a > 0 and deduce that

Solution: 

Applying the inversion formula, we get

Example 4.3.d(2): Find the Fourier sine transform of e-x, Hence show that

Solution: 

Changing 'x' to 'm' and 's' to 'x', we get

Example 44d(3): Find f(x) if its sine transform is

Hence, find

Solution: 

Integrating w.r.to x, we get

Example 4.3.d(4): Solve the integral equation

Solution: 

By inversion formula, we get

III.(e) Problems based on properties of F.C.T AND F.S.T.

Example 4.3.e(1): (i) Find the Fourier cosine transform of

(ii) Find the Fourier sine transform of

Solution: 

Solving (6) & (7), we get

Example 4.3.e(2): Find the Fourier sine and cosine transformations of xe-ax.

Solution:

Example 4.3.e(3): Find Fourier cosine transform of and hence find ]

Solution : See Example 4.3 a(13), Page No. 4.97

Example 4.3.e(4): Find the Fourier sine transform of e-ax and hence find the Fourier cosine transform of xe-ax.

Solution: We know that,

III (f) Problems based on Parseval's identity in F.S.T and F.C.T

Example 4.3.f(1): Evaluate : using transforms.

Solution :

Parseval's identity is

Note :

(1) Evaluate:

Solution : Step 1:

(2) Evaluate :

Solution :

Example 4.3.f(2): Using Fourier sine transform, prove that

Solution : Parseval's identity is

Solution: 

Example 4.3.f(3) : Using transform methods, evaluate

Solution : Parseval's identity is

Example 4.3.f(4): Using transform methods, evaluate where a > 0. 

Solution : Parseval's identity is

Example 4.3.f(5): Using Parseval's identity of the Fourier cosine transform, Evaluate

Solution :

Parseval's identity is

EXERCISE 4.1 [Fourier integral theorem

EXERCISE 4.2 [Fourier Transform]

1. Find the Fourier transform of the function.

EXERCISE 4.3 [FCT and FST]

I. 1. Find the Fourier cosine transform of e-4x. Deduce that

II. Find the Fourier Cosine Transform of

III. Find the Fourier Sine transform of

IV. Find the Fourier cosine and sine transforms of