A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter 3.2, Problem 12E
Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.
- Claim. Let o be the operation on
"Proof."
"Proof."
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a
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5. (a) State the Residue Theorem. Your answer should include all the conditions required
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(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
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ཙ
x sin(Tx)
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da.
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3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
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(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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(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)
Chapter 3 Solutions
A Transition to Advanced Mathematics
Ch. 3.1 - Let 3 and 6 be the sets of integer multiples of 3...Ch. 3.1 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 3.1 - Let ({a,b,c},o) be the group with the operation...Ch. 3.1 - (a)Prove that the function f:1824 given by f(x)=4x...Ch. 3.1 - Define f:1512 by f(x)=4x. Prove that f is a...Ch. 3.1 - Let (G,) and (H,*) be groups, i be the identity...Ch. 3.1 - Prob. 7ECh. 3.1 - Prob. 8ECh. 3.1 - Prove that the relation of isomorphism is an...Ch. 3.1 - Prob. 10E
Ch. 3.1 - Prove that if G is a group and H is a subgroup of...Ch. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 3.1 - Prob. 14ECh. 3.1 - Prob. 15ECh. 3.1 - Prob. 16ECh. 3.1 - Prob. 17ECh. 3.2 - (a)Show that any two groups of order 2 are...Ch. 3.2 - (a)Show that the function h: defined by h(x)=3x is...Ch. 3.2 - Let R be the equivalence relation on ({0}) given...Ch. 3.2 - Let (R,+,) be an integral domain. Prove that 0 has...Ch. 3.2 - Complete the proof of Theorem 6.5.5. That is,...Ch. 3.2 - Prob. 6ECh. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 8ECh. 3.2 - Prob. 9ECh. 3.2 - Use the method of proof of Cayley's Theorem to...Ch. 3.2 - Prob. 11ECh. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 13ECh. 3.2 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 3.2 - Prob. 15ECh. 3.2 - Let f:(A,)(B,*) and g:(B,*)(C,X) be OP maps. Prove...Ch. 3.2 - Prob. 17ECh. 3.2 - Let Conj: be the conjugate mapping for complex...Ch. 3.2 - Prove the remaining parts of Theorem 6.4.1.Ch. 3.3 - Let 3={3k:k}. Apply the Subring Test (Exercise...Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Prob. 6ECh. 3.3 - Use the definition of “divides” to explain (a) why...Ch. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Complete the proof that for every m,(m+,) is a...Ch. 3.3 - Define addition and multiplication on the set ...Ch. 3.3 - Prob. 12ECh. 3.3 - Let (R,+,) be a ring and a,b,R. Prove that b+(a)...Ch. 3.3 - Prove the remaining parts of Theorem 6.5.3: For...Ch. 3.3 - We define a subring of a ring in the same way we...Ch. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - If possible, give an example of a set A such that...Ch. 3.4 - Let A. Prove that if sup(A) exists, then...Ch. 3.4 - Let A and B be subsets of . Prove that if sup(A)...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - An alternate version of the Archimedean Principle...Ch. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.5 - Prob. 1ECh. 3.5 - Prob. 2ECh. 3.5 - Let A be a subset of . Prove that the set of all...Ch. 3.5 - Prob. 4ECh. 3.5 - Let be an associative operation on nonempty set A...Ch. 3.5 - Suppose that (A,*) is an algebraic system and * is...Ch. 3.5 - Let (A,o) be an algebra structure. An element lA...Ch. 3.5 - Let G be a group. Prove that if a2=e for all aG,...Ch. 3.5 - Give an example of an algebraic structure of order...Ch. 3.5 - Prove that an ordered field F is complete iff...Ch. 3.5 - Prove that every irrational number is "missing"...Ch. 3.5 - Find two upper bounds (if any exits) for each of...Ch. 3.5 - Prob. 13ECh. 3.5 - Prob. 14ECh. 3.5 - Prob. 15ECh. 3.5 - Let A and B be subsets of . Prove that if A is...Ch. 3.5 - Prob. 17ECh. 3.5 - Prob. 18ECh. 3.5 - Give an example of a set A for which both A and Ac...Ch. 3.5 - Prob. 20ECh. 3.5 - Prob. 21ECh. 3.5 - Prob. 22E
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- 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?arrow_forward(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_forward
- 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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