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 25.1, Problem 15E
To determine
The anti-derivative of the given function
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5. (a) State the Residue Theorem. Your answer should include all the conditions required
for the theorem to hold.
(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
anti-clockwise direction. Evaluate
に
dz.
You must check all of the conditions of any results that you use.
(5 marks)
(c) Evaluate
L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
x²+2x+5
da.
(11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
for L(y).
(1 mark)
(b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a
contour. Suppose there exists a finite real number M such that |f(z)| < M for
all z in the image of y. Prove that
<
||, f(z)dz| ≤ ML(y).
(3 marks)
(c) State and prove Liouville's theorem. You may use Cauchy's integral formula without
proof.
(d) Let R0. Let w € C. Let
(10 marks)
U = { z Є C : | z − w| < R} .
Let f UC be a holomorphic function such that
0 < |ƒ(w)| < |f(z)|
for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
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?
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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