Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Chapter 47, Problem 9AR
To determine
(a)
To solve problem for following given Equation.
To determine
(b)
To solve problem for following given Equation.
To determine
(c)
To solve problem for following given Equation.
To determine
(d)
To solve problem for following given Equation.
To determine
(e)
To solve problem for following given Equation.
To determine
(f)
To solve problem for following given Equation.
To determine
(g)
To solve problem for following given Equation.
To determine
(h)
To solve problem for following given Equation.
To determine
(i)
To solve problem for following given Equation.
To determine
(j)
To solve problem for following given Equation.
To determine
(k)
To solve problem for following given Equation.
To determine
(l)
To solve problem for following given Equation.
To determine
(m)
To solve problem for following given Equation.
To determine
(n)
To solve problem for following given Equation.
To determine
(o)
To solve problem for following given Equation.
To determine
(p)
To solve problem for following given Equation.
To determine
q
To solve problem for following given Equation.
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Students have asked these similar questions
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?
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