EBK MATHEMATICS FOR MACHINE TECHNOLOGY
7th Edition
ISBN: 9780100548169
Author: SMITH
Publisher: YUZU
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Chapter 51, Problem 15A
Solve the following exercises:
Find the value of
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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 51 Solutions
EBK MATHEMATICS FOR MACHINE TECHNOLOGY
Ch. 51 - Determine the values of 2, 3, and 4 if l is 113.Ch. 51 - Use a protractor to measure the angle to the...Ch. 51 - Express 191.5326 as degrees, minutes, and seconds....Ch. 51 - Cast iron 10 cm in diameter is turned in a lathe...Ch. 51 - Solve 4t7t2216=12t.Ch. 51 - If m m=5,p=2,and r=12 ,find m24p+3rmp+prmmr.Give...Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Identify each of the triangles 7 through 14 as...
Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Identify each of the triangles 7 through 14 as...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: In triangle ABC, BC...Ch. 51 - Solve the following exercises: In triangle EFG,...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: All dimensions are...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: Hole centrelines...Ch. 51 - Solve the following exercises: Find the value of...Ch. 51 - Solve the following exercises: ABDE,BC is an...Ch. 51 - Determine the answers to the following exercises...Ch. 51 - Determine the answers to the following exercises...Ch. 51 - Determine the answers to the following exercises...Ch. 51 - Determine the answers to the following exercises...
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