Transforms and its Applications: UNIT 4: Fourier Transform
Fourier Cosine Transform and its Inversion formula
Fourier Transform: Formula, Statement, Example Important Solved Problems with formula, steps, derivation and answer based on Fourier cosine transform and its inversion formula.
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
Now
define g(x) by g(x) = 
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).
=
Fc [g(x)]
=
Fc [f(x)] [ g(x) = f(x) for all x ≥ 0]
Hence,
by inversion formula for F.T, we have
Formula:
Problems based on Fourier cosine transform and its inversion formula
Example 1: Solve the integral equation 
and also show that 
.
Solution:
Example 2: Solve the integral equation
Solution:
Example 3: Find the Fourier cosine transform
of e‒|x| and deduce that 
.
Solution:
We know that,
Example 4: Find the Fourier cosine transform
of e-ax, a>0 and deduce that 
.
Solution:
We know that,
Applying
the inversion formula, we have
Transforms and its Applications: UNIT 4: Fourier Transform : Tag: Engineering mathematics, Maths : - Fourier Cosine Transform and its Inversion formula
Transforms and its Applications: UNIT 4: Fourier Transform
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