21.3. Evaluate the integral dx/[(x² + 1)(x² + 4)] by closing the contour (a) in the UHP and (b) in the LHP.
21.3. Evaluate the integral dx/[(x² + 1)(x² + 4)] by closing the contour (a) in the UHP and (b) in the LHP.
Chapter2: Functions And Their Graphs
Section2.3: Analyzing Graphs Of Functions
Problem 6ECP
Related questions
Question
100%
The question is about complex analysis and Calculus of Residues.
One of the 2 pictures is an example of a solution.
![TESTTUE of f(Z)=
21.3. Evaluate the integral f dx/[(x2 + 1)(x² + 4)] by closing the contour
(a) in the UHP and (b) in the LHP.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94dc01b6-0881-4fad-b8f8-3d9bb8d43f15%2Fae6c0a5e-b80a-4d09-8cce-05957df508ed%2Fy3zelwv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:TESTTUE of f(Z)=
21.3. Evaluate the integral f dx/[(x2 + 1)(x² + 4)] by closing the contour
(a) in the UHP and (b) in the LHP.
![Example 21.2.2. Let us now consider a more complicated integral:
x2 dx
(x2 + 1)(x² + 4)²
which turns into f, z2 dz/[(z² +1)(z² +4)²]. The poles in the UHP are at z = i and
= 2i. The former is a simple pole, and the latter is a pole of order 2. Thus,
22
1
Res[f(i)] = lim
|(z
i)
(z – i)(z + i)(z²+ 4)².
18i
22
1
Res[f(2i)] =
(2 – 1)! " dz (2 – 2i)².
22
d
lim
1)! z→2i dz
(z2 + 1)(z + 2i)² (z – 2i)².
d
lim
z→2i dz
[(22 + 1)(z + 2i)².
72i'
and
x² dx
5
2ni
(x2 + 1)(x² + 4)2
18i
72i
36
Closing the contour in the LHP would yield the same result as the reader is urged
to verify.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94dc01b6-0881-4fad-b8f8-3d9bb8d43f15%2Fae6c0a5e-b80a-4d09-8cce-05957df508ed%2Fcd6st7p_processed.png&w=3840&q=75)
Transcribed Image Text:Example 21.2.2. Let us now consider a more complicated integral:
x2 dx
(x2 + 1)(x² + 4)²
which turns into f, z2 dz/[(z² +1)(z² +4)²]. The poles in the UHP are at z = i and
= 2i. The former is a simple pole, and the latter is a pole of order 2. Thus,
22
1
Res[f(i)] = lim
|(z
i)
(z – i)(z + i)(z²+ 4)².
18i
22
1
Res[f(2i)] =
(2 – 1)! " dz (2 – 2i)².
22
d
lim
1)! z→2i dz
(z2 + 1)(z + 2i)² (z – 2i)².
d
lim
z→2i dz
[(22 + 1)(z + 2i)².
72i'
and
x² dx
5
2ni
(x2 + 1)(x² + 4)2
18i
72i
36
Closing the contour in the LHP would yield the same result as the reader is urged
to verify.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
