Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 85, Problem 26A
To determine
To express:
A hexadecimal number into a binary number.
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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)
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L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
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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 85 Solutions
Mathematics For Machine Technology
Ch. 85 - Prob. 1ACh. 85 - Prob. 2ACh. 85 - Prob. 3ACh. 85 - Prob. 4ACh. 85 - Prob. 5ACh. 85 - Prob. 6ACh. 85 - Prob. 7ACh. 85 - Prob. 8ACh. 85 - Prob. 9ACh. 85 - Prob. 10A
Ch. 85 - Prob. 11ACh. 85 - Prob. 12ACh. 85 - Prob. 13ACh. 85 - Prob. 14ACh. 85 - Prob. 15ACh. 85 - Prob. 16ACh. 85 - Prob. 17ACh. 85 - Prob. 18ACh. 85 - Prob. 19ACh. 85 - Prob. 20ACh. 85 - Prob. 21ACh. 85 - Prob. 22ACh. 85 - Prob. 23ACh. 85 - Prob. 24ACh. 85 - Prob. 25ACh. 85 - Prob. 26ACh. 85 - Prob. 27ACh. 85 - Prob. 28ACh. 85 - Prob. 29ACh. 85 - Prob. 30ACh. 85 - Prob. 31ACh. 85 - Prob. 32ACh. 85 - Prob. 33ACh. 85 - Prob. 34ACh. 85 - Prob. 35ACh. 85 - Prob. 36ACh. 85 - Prob. 37ACh. 85 - Prob. 38ACh. 85 - Prob. 39ACh. 85 - Prob. 40ACh. 85 - Prob. 41ACh. 85 - Prob. 42ACh. 85 - Prob. 43ACh. 85 - Prob. 44ACh. 85 - Prob. 45ACh. 85 - Prob. 46A
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