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 8AR
To determine
(a)
To solve problem for following given formulation.
To determine
(b)
To solve problem for following given formulation.
To determine
(c)
To solve problem for following given formulation.
To determine
(d)
To solve problem for following given formulation.
To determine
(e)
To solve problem for following given formulation problem.
To determine
(f)
To solve problem for following given formulation problem.
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Theorem: show that XCH) = M(E) M" (6) E +
t
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a
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x = ACE) x + g (t) + X (E) - E
5. (a) State the Residue Theorem. Your answer should include all the conditions required
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(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
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に
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(c) Evaluate
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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
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<
||, f(z)dz| ≤ ML(y).
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(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} .
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for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
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