Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Chapter 48, Problem 49A
Determine
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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)
(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?
Chapter 48 Solutions
Mathematics for Machine Technology
Ch. 48 - Refer to the Decimal-Inch Spur Gears Table under...Ch. 48 - If the cutting speed is 160 m/min and the...Ch. 48 - Solve the equation 28A2=(5+A)(2A)for A.Ch. 48 - Use the addition of equality to solve...Ch. 48 - If a=5.2,b=4.8,and c=7.25 ,what is the value of...Ch. 48 - Prob. 6ACh. 48 - Prob. 7ACh. 48 - Prob. 8ACh. 48 - Write the symbols for the following words. a....Ch. 48 - Express the following decimal degrees as degrees...
Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the following decimal degrees as degrees...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the decimal degrees as degrees, minutes,...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees and minutes as...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Express the following degrees, minutes, and...Ch. 48 - Determine 1.Ch. 48 - Determine 2.Ch. 48 - Determine 3.Ch. 48 - Determine 1+2+3.Ch. 48 - Determine 5.Ch. 48 - Determine 6.Ch. 48 - Determine 7+8+9Ch. 48 - Determine 1+2+3+4+5.Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Subtract the angles in each of the following...Ch. 48 - Multiply the angles in each of the following...Ch. 48 - Multiply the angles in each of the following...Ch. 48 - Multiply the angles in each of the following...Ch. 48 - Multiply the angles in each of the following...Ch. 48 - Multiply the angles in each of the following...Ch. 48 - In the figure shown, 1=2=42. Determine 3.Ch. 48 - IF x=3914,find 4.Ch. 48 - In the figure shown, 1=2=3=4=5=5403. Determine 6.Ch. 48 - Divide the angles in each of the following...Ch. 48 - Divide the angles in each of the following...Ch. 48 - Divide the angles in each of the following...Ch. 48 - Divide the angles in each of the following...Ch. 48 - Divide the angles in each of the following...Ch. 48 - Divide the angles in each of the following...Ch. 48 - Divide the angles in each of the following...Ch. 48 - The sum of the angles in figure equals shown...
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- (a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's integral formula for derivatives. Your answer should include all the conditions required for the results to hold. (8 marks) (b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at 0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate dz. You must check the conditions of any results you use. (d) Let U C. Calculate Liz-1ym dz, (z - 1) 10 (5 marks) where 2 is the same as the previous part. You must check the conditions of any results you use. (4 marks)arrow_forward(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it means for this singularity to be a pole of order k. (2 marks) (b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given by 1 res (f, w): = Z dk (k-1)! >wdzk−1 lim - [(z — w)* f(z)] . (5 marks) (c) Using the previous part, find the singularity of the function 9(z) = COS(πZ) e² (z - 1)²' classify it and calculate its residue. (5 marks) (d) Let g(x)=sin(211). Find the residue of g at z = 1. (3 marks) (e) Classify the singularity of cot(z) h(z) = Z at the origin. (5 marks)arrow_forward1. 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_forward
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