Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)
11th Edition
ISBN: 9780137554843
Author: Allyn Washington, Richard Evans
Publisher: PEARSON+
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Chapter 25, Problem 22RE
To determine
The value of the integral
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3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
(a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's
integral formula for derivatives. Your answer should include all the conditions
required for the results to hold.
(8 marks)
(b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at
0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate
dz.
You must check the conditions of any results you use.
(d) Let U C. Calculate
Liz-1ym dz,
(z - 1) 10
(5 marks)
where 2 is the same as the previous part. You must check the conditions of any
results you use.
(4 marks)
(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it
means for this singularity to be a pole of order k.
(2 marks)
(b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given
by
1
res (f, w):
=
Z
dk
(k-1)! >wdzk−1
lim
-
[(z — w)* f(z)] .
(5 marks)
(c) Using the previous part, find the singularity of the function
9(z) =
COS(πZ)
e² (z - 1)²'
classify it and calculate its residue.
(5 marks)
(d) Let g(x)=sin(211). Find the residue of g at z = 1.
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Chapter 25 Solutions
Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)
Ch. 25.1 - Find an antiderivative of x3 + 4x.
Ch. 25.1 - Prob. 2PECh. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - In Exercises 5–12, determine the value of a that...Ch. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8E
Ch. 25.1 - Prob. 9ECh. 25.1 - In Exercises 5–12, determine the value of a that...Ch. 25.1 - Prob. 11ECh. 25.1 - Prob. 12ECh. 25.1 - Prob. 13ECh. 25.1 - Prob. 14ECh. 25.1 - Prob. 15ECh. 25.1 - Prob. 16ECh. 25.1 - Prob. 17ECh. 25.1 - Prob. 18ECh. 25.1 - Prob. 19ECh. 25.1 - In Exercises 13–40, find antiderivatives of the...Ch. 25.1 - Prob. 21ECh. 25.1 - Prob. 22ECh. 25.1 - Prob. 23ECh. 25.1 - Prob. 24ECh. 25.1 - Prob. 25ECh. 25.1 - Prob. 26ECh. 25.1 - Prob. 27ECh. 25.1 - Prob. 28ECh. 25.1 - Prob. 29ECh. 25.1 - Prob. 30ECh. 25.1 - Prob. 31ECh. 25.1 - Prob. 32ECh. 25.1 - Prob. 33ECh. 25.1 - Prob. 34ECh. 25.1 - Prob. 35ECh. 25.1 - Prob. 36ECh. 25.1 - Prob. 37ECh. 25.1 - Prob. 38ECh. 25.1 - Prob. 39ECh. 25.1 - Prob. 40ECh. 25.1 - Prob. 41ECh. 25.1 - Prob. 42ECh. 25.2 - Integrate: .
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - In Exercise 5–36, integrate each of the given...Ch. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - In Exercise 5–36, integrate each of the given...Ch. 25.2 - In Exercise 5–36, integrate each of the given...Ch. 25.2 - Prob. 11ECh. 25.2 - Prob. 12ECh. 25.2 - Prob. 13ECh. 25.2 - Prob. 14ECh. 25.2 - Prob. 15ECh. 25.2 - Prob. 16ECh. 25.2 - Prob. 17ECh. 25.2 - Prob. 18ECh. 25.2 - Prob. 19ECh. 25.2 - Prob. 20ECh. 25.2 - Prob. 21ECh. 25.2 - In Exercises 5–36, integrate each of the given...Ch. 25.2 - Prob. 23ECh. 25.2 - Prob. 24ECh. 25.2 - Prob. 25ECh. 25.2 - Prob. 26ECh. 25.2 - Prob. 27ECh. 25.2 - Prob. 28ECh. 25.2 - Prob. 29ECh. 25.2 - Prob. 30ECh. 25.2 - Prob. 31ECh. 25.2 - Prob. 32ECh. 25.2 - Prob. 33ECh. 25.2 - Prob. 34ECh. 25.2 - Prob. 35ECh. 25.2 - Prob. 36ECh. 25.2 - Prob. 37ECh. 25.2 - Prob. 38ECh. 25.2 - Prob. 39ECh. 25.2 - Prob. 40ECh. 25.2 - Prob. 41ECh. 25.2 - Prob. 42ECh. 25.2 - Prob. 43ECh. 25.2 - Prob. 44ECh. 25.2 - Prob. 45ECh. 25.2 - Prob. 46ECh. 25.2 - Prob. 47ECh. 25.2 - Prob. 48ECh. 25.2 - Prob. 49ECh. 25.2 - Prob. 50ECh. 25.2 - Prob. 51ECh. 25.2 - Prob. 52ECh. 25.2 - Prob. 53ECh. 25.2 - Prob. 54ECh. 25.2 - In Exercises 41–62, solve the given problems. In...Ch. 25.2 - Prob. 56ECh. 25.2 - Prob. 57ECh. 25.2 - Prob. 58ECh. 25.2 - Prob. 59ECh. 25.2 - Prob. 60ECh. 25.2 - Prob. 61ECh. 25.2 - Prob. 62ECh. 25.3 - Prob. 1PECh. 25.3 - Prob. 2PECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25.3 - Prob. 7ECh. 25.3 - Prob. 8ECh. 25.3 - Prob. 9ECh. 25.3 - Prob. 10ECh. 25.3 - Prob. 11ECh. 25.3 - Prob. 12ECh. 25.3 - Prob. 13ECh. 25.3 - Prob. 14ECh. 25.3 - Prob. 15ECh. 25.3 - Prob. 16ECh. 25.3 - Prob. 17ECh. 25.3 - Prob. 18ECh. 25.3 - Prob. 19ECh. 25.3 - Prob. 20ECh. 25.3 - Prob. 21ECh. 25.3 - Prob. 22ECh. 25.3 - Prob. 23ECh. 25.3 - In Exercises 15–24, find the exact area under the...Ch. 25.3 - Prob. 25ECh. 25.3 - Prob. 26ECh. 25.3 - Prob. 27ECh. 25.3 - Prob. 28ECh. 25.4 -
Evaluate: .
Ch. 25.4 - Prob. 2PECh. 25.4 - Prob. 1ECh. 25.4 - Prob. 2ECh. 25.4 - Prob. 3ECh. 25.4 - Prob. 4ECh. 25.4 - Prob. 5ECh. 25.4 - Prob. 6ECh. 25.4 - Prob. 7ECh. 25.4 - Prob. 8ECh. 25.4 - Prob. 9ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 12ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 15ECh. 25.4 - Prob. 16ECh. 25.4 - Prob. 17ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 20ECh. 25.4 - Prob. 21ECh. 25.4 - Prob. 22ECh. 25.4 - Prob. 23ECh. 25.4 - Prob. 24ECh. 25.4 - Prob. 25ECh. 25.4 - Prob. 26ECh. 25.4 - Prob. 27ECh. 25.4 - Prob. 28ECh. 25.4 - Prob. 29ECh. 25.4 - Prob. 30ECh. 25.4 - Prob. 31ECh. 25.4 - Prob. 32ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 34ECh. 25.4 - In Exercises 35–54, solve the given problems.
35....Ch. 25.4 - Prob. 36ECh. 25.4 - In Exercises 35–54, solve the given problems.
37....Ch. 25.4 - Prob. 38ECh. 25.4 - Prob. 39ECh. 25.4 - Prob. 40ECh. 25.4 - Prob. 41ECh. 25.4 - Prob. 42ECh. 25.4 - Prob. 43ECh. 25.4 - Prob. 44ECh. 25.4 - Prob. 45ECh. 25.4 - Prob. 46ECh. 25.4 - Prob. 47ECh. 25.4 - Prob. 48ECh. 25.4 - Prob. 49ECh. 25.4 - Prob. 50ECh. 25.4 - Prob. 51ECh. 25.4 - Prob. 52ECh. 25.4 - In finding the average electron energy in a metal...Ch. 25.4 - Prob. 54ECh. 25.5 - Prob. 1PECh. 25.5 - Prob. 1ECh. 25.5 - Prob. 2ECh. 25.5 - Prob. 3ECh. 25.5 - Prob. 4ECh. 25.5 - Prob. 5ECh. 25.5 - Prob. 6ECh. 25.5 - Prob. 7ECh. 25.5 - Prob. 8ECh. 25.5 - Prob. 9ECh. 25.5 - Prob. 10ECh. 25.5 - Prob. 11ECh. 25.5 - Prob. 12ECh. 25.5 - Prob. 13ECh. 25.5 - Prob. 14ECh. 25.5 - Prob. 15ECh. 25.5 - Prob. 16ECh. 25.5 - Prob. 17ECh. 25.5 - Prob. 18ECh. 25.5 - Prob. 19ECh. 25.5 - Prob. 20ECh. 25.5 - Prob. 21ECh. 25.5 - Prob. 22ECh. 25.6 - Prob. 1PECh. 25.6 - Prob. 1ECh. 25.6 - Prob. 2ECh. 25.6 - Prob. 3ECh. 25.6 - Prob. 4ECh. 25.6 - Prob. 5ECh. 25.6 - Prob. 6ECh. 25.6 - Prob. 7ECh. 25.6 - Prob. 8ECh. 25.6 - Prob. 9ECh. 25.6 - Prob. 10ECh. 25.6 - Prob. 11ECh. 25.6 - Prob. 12ECh. 25.6 - Prob. 13ECh. 25.6 - Prob. 14ECh. 25.6 - Prob. 15ECh. 25.6 - Prob. 16ECh. 25.6 - Prob. 17ECh. 25.6 - Prob. 18ECh. 25 - Prob. 1RECh. 25 - Determine each of the following as being either...Ch. 25 - Prob. 3RECh. 25 - Prob. 4RECh. 25 - Prob. 5RECh. 25 - Prob. 6RECh. 25 - Prob. 7RECh. 25 - Prob. 8RECh. 25 - Prob. 9RECh. 25 - Prob. 10RECh. 25 - Prob. 11RECh. 25 - Prob. 12RECh. 25 - Prob. 13RECh. 25 - Prob. 14RECh. 25 - Prob. 15RECh. 25 - Prob. 16RECh. 25 - Prob. 17RECh. 25 - Prob. 18RECh. 25 - Prob. 19RECh. 25 - Prob. 20RECh. 25 - Prob. 21RECh. 25 - Prob. 22RECh. 25 - Prob. 23RECh. 25 - Prob. 24RECh. 25 - Prob. 25RECh. 25 - Prob. 26RECh. 25 - Prob. 27RECh. 25 - Prob. 28RECh. 25 - Prob. 29RECh. 25 - Prob. 30RECh. 25 - Prob. 31RECh. 25 - Prob. 32RECh. 25 - Prob. 33RECh. 25 - Prob. 34RECh. 25 - Prob. 35RECh. 25 - Prob. 36RECh. 25 - Prob. 37RECh. 25 - Prob. 38RECh. 25 - Prob. 39RECh. 25 - Prob. 40RECh. 25 - Prob. 41RECh. 25 - Prob. 42RECh. 25 - Prob. 43RECh. 25 - Prob. 44RECh. 25 - Prob. 45RECh. 25 - Prob. 46RECh. 25 - Prob. 47RECh. 25 - Prob. 48RECh. 25 - Prob. 49RECh. 25 - Prob. 50RECh. 25 - Prob. 51RECh. 25 - Prob. 52RECh. 25 - Prob. 53RECh. 25 - Prob. 54RECh. 25 - Prob. 55RECh. 25 - Prob. 56RECh. 25 - Prob. 57RECh. 25 - Prob. 58RECh. 25 - Prob. 59RECh. 25 - Prob. 60RECh. 25 - Prob. 61RECh. 25 - Prob. 62RECh. 25 - Prob. 63RECh. 25 - Prob. 64RECh. 25 - Prob. 65RECh. 25 - Prob. 66RECh. 25 - Prob. 67RECh. 25 - Prob. 68RECh. 25 - Prob. 1PTCh. 25 - Prob. 2PTCh. 25 - Prob. 3PTCh. 25 - Prob. 4PTCh. 25 - Prob. 5PTCh. 25 - Prob. 6PTCh. 25 - Prob. 7PT
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