Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Chapter 53, Problem 3A
To determine
To find the angle in given triangle in the image.
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Theorem: show that XCH) = M(E) M" (6) E +
t
Mcfic
S
a
Solution of ODE
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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
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)
Chapter 53 Solutions
Mathematics for Machine Technology
Ch. 53 - Determine the length of a. Round the answer to 1...Ch. 53 - Prob. 2ACh. 53 - Prob. 3ACh. 53 - Prob. 4ACh. 53 - Prob. 5ACh. 53 - Prob. 6ACh. 53 - Name each of the parts of circles for the...Ch. 53 - Name each of the parts of circles for the...Ch. 53 - Name each of the parts of circles for the...Ch. 53 - Name each of the parts of circles for the...
Ch. 53 - Prob. 11ACh. 53 - Circumference Formula Use C= or C=2r where C=...Ch. 53 - Prob. 13ACh. 53 - Circumference Formula Use C= or C=2r where C=...Ch. 53 - Solve the following exercises based on Principles...Ch. 53 - Solve the following exercises based on Principles...Ch. 53 - Prob. 17ACh. 53 - Solve the following exercises based on Principles...Ch. 53 - Prob. 19ACh. 53 - Solve the following exercises based on Principles...Ch. 53 - Prob. 21ACh. 53 - Solve the following exercises based on Principles...Ch. 53 - Solve the following exercises based on Principles...Ch. 53 - Solve the following exercises based on Principles...Ch. 53 - Prob. 25ACh. 53 - Prob. 26ACh. 53 - Prob. 27ACh. 53 - Solve the following exercises based on Principles...Ch. 53 - Solve the following exercises based on Principles...Ch. 53 - Solve the following exercises based on Principles...Ch. 53 - Prob. 31ACh. 53 - Solve the following exercises based on Principles...
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- 1. Let z = x+iy with x, y Є R. Let f(z) = u(x, y) + iv(x, y) where u(x, y), v(x, y): R² → R. (a) Suppose that f is complex differentiable. State the Cauchy-Riemann equations satisfied by the functions u(x, y) and v(x,y). (b) State what it means for the function (2 mark) u(x, y): R² → R to be a harmonic function. (3 marks) (c) Show that the function u(x, y) = 3x²y - y³ +2 is harmonic. (d) Find a harmonic conjugate of u(x, y). (6 marks) (9 marks)arrow_forwardLet A be a vector space with basis 1, a, b. Which (if any) of the following rules turn A into an algebra? (You may assume that 1 is a unit.) (i) a² = a, b² = ab = ba = 0. (ii) a²=b, b² = ab = ba = 0. (iii) a²=b, b² = b, ab = ba = 0.arrow_forwardNo chatgpt pls will upvotearrow_forward
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