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Fourier Transform Assignments

1. The document is an assignment sheet from a Mathematics III course covering Fourier transforms. It contains 23 questions involving expressing functions as Fourier integrals and evaluating integrals using Fourier transforms and Parseval's identity. 2. Questions 1-18 involve expressing functions as Fourier integrals or Fourier sine/cosine transforms and evaluating related integrals. Questions 19-23 involve using Parseval's identity to evaluate definite integrals. 3. The assignment is due on December 2nd, 2022 and covers material related to Fourier transforms, including expressing functions as Fourier integrals and using properties of Fourier transforms to evaluate definite integrals.

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201 views2 pages

Fourier Transform Assignments

1. The document is an assignment sheet from a Mathematics III course covering Fourier transforms. It contains 23 questions involving expressing functions as Fourier integrals and evaluating integrals using Fourier transforms and Parseval's identity. 2. Questions 1-18 involve expressing functions as Fourier integrals or Fourier sine/cosine transforms and evaluating related integrals. Questions 19-23 involve using Parseval's identity to evaluate definite integrals. 3. The assignment is due on December 2nd, 2022 and covers material related to Fourier transforms, including expressing functions as Fourier integrals and using properties of Fourier transforms to evaluate definite integrals.

Uploaded by

Aditya
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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FORM NO.

F / CDTE / 16
REV. NO. ISSUE DATE 00, 01-07-2018

DEPARTMENT OF COMPUTER
ASSIGNMENT -III
SCIENCE & ENGG.

ACADEMIC YEAR 2022-2023 COURSE TEACHER I.M.PALKAR


COURSE NAME MATH - III CLASS S.E.
SEM. III DATE 02/12/2022

ASSIGNMENT NO. 03 TOTAL MARKS:05 M

Q.
Question Description
No.

1 1 ;0 ≤ 𝑥 ≤ 𝜋 ∞ 1−𝑐𝑜𝑠𝜆𝜋
Express 𝑓(𝑡) = { as a Fourier sine integral and hence deduce that ∫0 𝑠𝑖𝑛𝜋𝜆 𝑑𝜆
0 ;𝑥 > 𝜋 𝜆

2 𝑐𝑜𝑠𝑥 ; |𝑥| < 𝜋


Express the function as Fourier integral 𝑓(𝑥) = {
0 ; |𝑥| > 𝜋

3 1 − 𝑥 2 ; |𝑥| ≤ 1
Find the Fourier transform of 𝑓(𝑥) = {
0 ; |𝑥| > 1

1 − 𝑥 2 ; |𝑥| ≤ 1
Find the Fourier transform of 𝑓(𝑥) = { ; Hence Prove that
0 ; |𝑥| > 1
4
∞
(𝑥𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥) 𝑥 −3𝜋
∫ 𝑐𝑜𝑠 𝑑𝑥 =
0 𝑥3 2 16

5 Find Fourier sine transform of 𝑓(𝑥) = 5𝑒 −2𝑥 + 2𝑒 −5𝑥

𝑠𝑖𝑛𝑥 ; 0 ≤ 𝑥 ≤ 𝜋
Express the function 𝑓(𝑥) = { as a Fourier sine integral and Evaluate that
6 0 ;𝑥 > 𝜋
∞ 𝑠𝑖𝑛𝜆𝑥 𝑠𝑖𝑛𝜆𝜋
∫0 𝑑𝜆
1−𝜆2

0 ; 𝑥 < −𝑎
7
Express the function as Fourier integral 𝑓(𝑥) = {1 ; −𝑎 < 𝑥 < 𝑎
0 ;𝑥 > 𝑎
𝜋
8 𝑐𝑜𝑠𝑥 ; |𝑥| < 2
Express the function as Fourier integral 𝑓(𝑥) = { 𝜋
0 ; |𝑥| > 2

9 𝜋
Express the function as Fourier sine integral 𝑓(𝑥) = 𝑒 −𝑥 ; 𝑥 ≥ 0
2

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10 𝑥3 ; 0 ≤ 𝑥 ≤ 1
Find the Fourier cosine and sine transform of 𝑓(𝑥) = {
0 ;𝑥 > 0

4𝑥 ;0 < 𝑥 < 1
11
Find Fourier sine transform of 𝑓(𝑥) = { 4 − 𝑥 ; 1 < 𝑥 < 4
0 ;𝑥 < 4

𝑥 , 0<𝑥<1
12 Find the Fourier Cosine transform of 𝑓(𝑥) = {2 − 𝑥, 1 < 𝑥 < 2
0, 𝑥 > 2

13 ∞ 𝑥𝑠𝑖𝑛𝑚𝑥 𝜋𝑒 −𝑚
Find the Fourier sine transform of 𝑒 −|𝑥| , and hence Show that ∮0 𝑑𝑥 = ,𝑚 > 0
1+𝑥 2 2

14 ∞ 𝑐𝑜𝑠𝜆𝑥
Find the Fourier cosine transform of 𝑒 −𝑎𝑥 Hence Evaluate ∫0 𝑑𝑥
𝑥 2 +𝑎2

𝜋
15 ∞ 𝑠𝑖𝑛𝜆𝜋 𝑠𝑖𝑛𝜆𝑥 𝑠𝑖𝑛𝑥 ; 0 ≤ 𝑥 ≤ 𝜋
Using the Fourier integral representations, show that ∫0 𝑑𝜆 = {2
1−𝜆3
0 ;𝑥 ≥ 𝜋

16 ∞ 𝑤𝑠𝑖𝑛𝑥𝑤 𝜋
Using the Fourier integral representations, show that ∫0 𝑑𝑤 = 2 𝑒 −𝑥 ; 𝑥 ≥ 0
1+𝑤 2

𝜋
17 ∞ 1−𝑐𝑜𝑠𝜆𝜋 ;0 ≤ 𝑥 ≤ 𝜋
Using the Fourier integral representations, show that ∫0 𝑠𝑖𝑛𝜆𝑥𝑑𝜆 = { 2
𝜆
0 ;𝑥 ≥ 𝜋

2
18 ∞ 𝑠𝑖𝑛𝑎𝑡 𝜋 1−𝑒 −𝑎
Using Parseval’s identity for cosine transform Prove that ∫0 𝑑𝑡 = 2 ( )
𝑡(𝑎2 +𝑡 2 ) 𝑎2

19 ∞ 𝑡2 𝜋
Using Parseval’s identity, Prove that ∫0 (𝑡 2 +1)2
𝑑𝑡 = 4

20 ∞ 𝑑𝑥
Using Parseval’s identity for cosine transform evaluate ∫0 (𝑥 2 +𝑎2 )(𝑥 2 +𝑏 2 )

21 ∞ sin2 𝑥
Using Parseval’s identity, Evaluate ∫0 𝑥2

22 ∞ 𝑑𝑡
Using Parseval’s identity, Evaluate ∫0 (𝑡 2 +1)2

23 ∞ 𝑡 2 𝑑𝑡
Using Parseval’s identity, Evaluate ∫0 (4+𝑡 2 )(9+𝑡 2 )

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