EBK MATHEMATICS FOR MACHINE TECHNOLOGY
7th Edition
ISBN: 9780100548169
Author: SMITH
Publisher: YUZU
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Chapter 47, Problem 1AR
To determine
(a)
To find out the algebraic expression for the given statement.
Expert Solution
Answer to Problem 1AR
Explanation of Solution
Given information:
The sum of
Calculation:
Its known that from the given problem use ofsum of
Apply the rule of addition property of equality, the result is
Now apply the rule of subtraction property of equality, the result is
To determine
(b)
To find out the algebraic expression for the given statement.
Expert Solution
Answer to Problem 1AR
Explanation of Solution
Given information:
The multiplication of
Calculation:
It's known that from the given problem
The multiplication of
Apply the multiplication property, the result is
Now apply the division property, the result is
To determine
(c)
To find out the algebraic expression for the given statement.
Expert Solution
Answer to Problem 1AR
Explanation of Solution
Given information:
Twice
Calculation:
Its known that from the given problem
Twice
Mathematically it can be written as
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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.
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(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
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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
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(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
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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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