Contemporary Mathematics for Business & Consumers
8th Edition
ISBN: 9781305886803
Author: Brechner
Publisher: Cengage
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Chapter 19.II, Problem 6TIE
To determine
To calculate: The value of regular refund if a property policy has an annual premium of
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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 19 Solutions
Contemporary Mathematics for Business & Consumers
Ch. 19.I - Prob. 1TIECh. 19.I - Prob. 2TIECh. 19.I - Prob. 3TIECh. 19.I - Calculate the annual, semiannual, quarterly, and...Ch. 19.I - Calculate the annual. semiannual, quarterly, and...Ch. 19.I - Prob. 3RECh. 19.I - Calculate the annual. semiannual, quarterly, and...Ch. 19.I - Prob. 5RECh. 19.I - Calculate the annual. semiannual, quarterly, and...Ch. 19.I - Calculate the annual. semiannual, quarterly, and...
Ch. 19.I - Calculate the annual. semiannual, quarterly, and...Ch. 19.I - Prob. 9RECh. 19.I - Calculate the value of the nonforfeiture options...Ch. 19.I - Prob. 11RECh. 19.I - Prob. 12RECh. 19.I - Calculate the value of the nonforfeiture options...Ch. 19.I - Calculate the value of the nonforfeiture options...Ch. 19.I - Leroy Kirk is 35 years old and is interested in...Ch. 19.I - 16. Rene Boyer, age 27. wants to purchase a 5-year...Ch. 19.I - Carmen Gutierrez purchased a $75,000, 20-payment...Ch. 19.I - 18. Alex Baron is evaluating his life insurance...Ch. 19.I - Richard Ryan is evaluating his life insurance...Ch. 19.I - BUSINESS DECISION: THE CONSULTATION
20. Tina...Ch. 19.II - You are the insurance agent for Diamond...Ch. 19.II - Prob. 5TIECh. 19.II - Prob. 6TIECh. 19.II - Prob. 7TIECh. 19.II - Prob. 8TIECh. 19.II - Prob. 1RECh. 19.II - Prob. 2RECh. 19.II - Prob. 3RECh. 19.II - Calculate the building, contents, and total...Ch. 19.II - Prob. 5RECh. 19.II - Prob. 6RECh. 19.II - Prob. 7RECh. 19.II - Prob. 8RECh. 19.II - Prob. 9RECh. 19.II - Calculate the short-term premium and refund for...Ch. 19.II - Calculate the short-term premium and refund for...Ch. 19.II - Calculate the short-term premium and refund for...Ch. 19.II - Prob. 13RECh. 19.II - Prob. 14RECh. 19.II - Prob. 15RECh. 19.II - Calculate the amount to be paid by the insurance...Ch. 19.II - Prob. 17RECh. 19.II - Calculate the amount to be paid by the insurance...Ch. 19.II - Prob. 19RECh. 19.II - You are the insurance agent for Castle Mountain...Ch. 19.II - A property insurance policy has an annual premium...Ch. 19.II - 22. Insignia Enterprises has a property insurance...Ch. 19.II - Prob. 23RECh. 19.II - BUSINESS DECISION: BUSINESS INTERRUPTION INSURANCE...Ch. 19.III - Jeff Wasserman, owner of High Performance Racing...Ch. 19.III - Prob. 10TIECh. 19.III - Prob. 1RECh. 19.III - Prob. 2RECh. 19.III - Prob. 3RECh. 19.III - As an insurance agent, calculate the annual...Ch. 19.III - Prob. 5RECh. 19.III - Prob. 6RECh. 19.III - Prob. 7RECh. 19.III - As an insurance agent, calculate the annual...Ch. 19.III - 9. Rick Clinton wants to purchase an automobile...Ch. 19.III -
10. Howard Marshall’s Corvette was hit by a palm...Ch. 19.III - Ben Hoffman has motor vehicle liability insurance...Ch. 19.III - BUSINESS DECISION: INSURING THE FLEET
12. The...Ch. 19 - A mechanism for reducing financial risk and...Ch. 19 - 2. The amount of protection provided by an...Ch. 19 - Prob. 3CRCh. 19 - Prob. 4CRCh. 19 - Prob. 5CRCh. 19 - Prob. 6CRCh. 19 - Prob. 7CRCh. 19 - Prob. 8CRCh. 19 - The premium charged when a policy is canceled by...Ch. 19 - The clause in a property insurance policy...Ch. 19 - Prob. 11CRCh. 19 - Prob. 12CRCh. 19 - Prob. 13CRCh. 19 - Prob. 14CRCh. 19 - Prob. 1ATCh. 19 - Calculate the annual, semiannual, quarterly, and...Ch. 19 - Calculate the annual, semiannual, quarterly, and...Ch. 19 - Calculate the annual, semiannual, quarterly, and...Ch. 19 - Prob. 5ATCh. 19 - Prob. 6ATCh. 19 - Prob. 7ATCh. 19 - 8. Mary Hall purchased a $45,000 20-year endowment...Ch. 19 - Prob. 9ATCh. 19 - Calculate the building, contents, and total...Ch. 19 - Prob. 11ATCh. 19 - Prob. 12ATCh. 19 - Prob. 13ATCh. 19 - Calculate the short-term premium and refund for...Ch. 19 - Prob. 15ATCh. 19 - Calculate the amount to be paid by the insurance...Ch. 19 - Prob. 17ATCh. 19 - Prob. 18ATCh. 19 - Prob. 19ATCh. 19 - Prob. 20ATCh. 19 - Prob. 21ATCh. 19 - Prob. 22ATCh. 19 - Prob. 23ATCh. 19 - Prob. 24AT
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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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