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Mathematics (maths) - Fourier Transforms : Important Questions and Answers: Fourier Transforms

**FOURIER TRANSFORMS**

**PART
–A**

**1. State Fourier integral theorem.**

If f(x) is piece-wise continuously differentiable and absolutely integrable in (- ¥, ¥) then

This
is known as Fourier integral theorem or Fourier integral formula.

**2. Define Fourier transform pair (or)
Define Fourier transform and its inverse transform.**

The complex (or infinite) Fourier transform of f(x)
is given by

Then the function f(x) is the inverse Fourier
Transform of F(s) and is given by

its also called Fourier Transform
Pairs.

**3. ****Show
that f(x) = 1, 0 < x < **¥**
cannot be represented by a Fourier integral.**

**4. State and prove the linear property
of FT.**

**5. State and prove the Shifting property
of FT. **

**Stt:**

**6. State and prove the Change of scale
property of FT. **

**Stt:**

**8. State and prove the Modulation
property of FT. (OR) If Fourier transform of f(x) is F(s).**

**Prove that the Fourier transform of ***f
*(*x*)cos* ax ***is**

**9. What is meant by self-reciprocal with
respect to FT?**

If
the Fourier transform of *f* (*x*)is
obtained just by replacing x by s, then *f*
(*x*)is called

self-reciprocal with respect to FT.

**12. Define Fourier cosine transform
(FCT) pair.**

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

**13.
****Find the Fourier Cosine transform
of f(x) =**

**14. Find the Fourier Cosine transform of
***e*^{-}^{ax}** , a > 0.**

Given *f*
(*x*)=*e*^{-}^{ax}

^{ }

**15. Find the Fourier Cosine transform of
***e*^{-}^{x}**
.**

We know that

**16. Define Fourier sine transform (FST)
pair.**

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

**17.
****Find the Fourier Sine transform of ***e*^{-}^{3x}**
.**

**18.
****Find the Fourier Sine transform of
f(x)= ***e*^{-}^{x}**
.**

**19. Find the Fourier Sine transform of **3*e*^{-}^{2}** **^{x}**
.**

Let *f* (*x*)=3*e*^{-}^{2} ^{x}

^{ }

**20.
****Find the Fourier Sine transform of
1/x.**

We know that

**21.
****State the Convolution theorem on
Fourier transform.**

**22.State**
**the Parseval’s formula or identity**

If *F s* is the Fourier transform of

**PART
B**

**1. State and prove the convolution
theorem for Fourier Transforms**.

Statement:

**PROOF**: By convolution
of two functions:

F[( *f* **g*
)(x)]=F(s)G(s)

3. **Show that** *e ^{x2/2} *

**Solution**:

Fourier transform:

**6. Find the Fourier cosine transform of ***e*^{-}^{a}^{2}**
**^{x}^{2}

**Solution**:

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