Contemporary Mathematics for Business & Consumers
8th Edition
ISBN: 9781305585447
Author: Robert Brechner, Geroge Bergeman
Publisher: Cengage Learning
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Chapter 12, Problem 5AT
To determine
To calculate: The present value of ordinary annuities where annuity payment is
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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?
(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)
(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)
Chapter 12 Solutions
Contemporary Mathematics for Business & Consumers
Ch. 12.I - Freeport Bank is paying 8% interest compounded...Ch. 12.I - Vista Savings Loan is paying 6% interest...Ch. 12.I - Katrina Byrd invested $250 at the end of every...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...
Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Use Table 12-1 to calculate the future value of...Ch. 12.I - Solve the following exercises by using Table 12-1....Ch. 12.I - Solve the following exercises by using Table 12-1....Ch. 12.I - Solve the following exercises by using Table...Ch. 12.I - Solve the following exercises by using Table 12-1....Ch. 12.I - Solve the following exercises by using Table...Ch. 12.I - Solve the following exercises by using formulas....Ch. 12.I - Solve the following exercises by using...Ch. 12.I - Solve the following exercises by using formulas....Ch. 12.I - Annuities Due Annuity Payment Time Nominal...Ch. 12.I - Annuities...Ch. 12.I - Annuities Due Annuity Payment Time Nominal...Ch. 12.I - To establish a "rainy day" cash reserve account....Ch. 12.I - 23. As a part of his retirement planning strategy....Ch. 12.I - Hi-Tech Hardware has been in business for a few...Ch. 12.II - Prob. 4TIECh. 12.II - Prob. 5TIECh. 12.II - Prob. 6TIECh. 12.II - Use Table 12-2 to calculate the present value of...Ch. 12.II - Prob. 2RECh. 12.II - Prob. 3RECh. 12.II - Prob. 4RECh. 12.II - Prob. 5RECh. 12.II - Prob. 6RECh. 12.II - Prob. 7RECh. 12.II - Prob. 8RECh. 12.II - Prob. 9RECh. 12.II - Prob. 10RECh. 12.II - Prob. 11RECh. 12.II - Solve the following exercises by using Table...Ch. 12.II - Solve the following exercises by using Table 12-2....Ch. 12.II - Prob. 14RECh. 12.II - Solve the following exercises by using Table...Ch. 12.II - Solve the following exercises by using Table...Ch. 12.II - Prob. 17RECh. 12.II - Prob. 18RECh. 12.II - Prob. 19RECh. 12.II - Prob. 20RECh. 12.II - Prob. 21RECh. 12.II - Prob. 22RECh. 12.II - As part of an inheritance. Joan Townsend will...Ch. 12.II - Norm Legend has been awarded a scholarship from...Ch. 12.III - Prob. 7TIECh. 12.III - Prob. 8TIECh. 12.III - Prob. 9TIECh. 12.III - Apex Manufacturing recently purchased a new...Ch. 12.III - Prob. 1RECh. 12.III - Prob. 2RECh. 12.III - Prob. 3RECh. 12.III - Prob. 4RECh. 12.III - Prob. 5RECh. 12.III - Prob. 6RECh. 12.III - You have just been hired as a loan officer at the...Ch. 12.III - Prob. 8RECh. 12.III - Prob. 9RECh. 12.III - Loan Payment Term of Nominal Present...Ch. 12.III - Prob. 11RECh. 12.III - Solve the following exercises by using tables.
12....Ch. 12.III - Solve the following exercises by using tables.
13....Ch. 12.III - Solve the following exercises by using tables....Ch. 12.III - Solve the following exercises by using tables.
15....Ch. 12.III - Solve the following exercises by using the sinking...Ch. 12.III - Prob. 17RECh. 12.III - Prob. 18RECh. 12.III - Prob. 19RECh. 12.III - Prob. 20RECh. 12.III - Prob. 21RECh. 12.III - Prob. 22RECh. 12.III - Randy Scott purchased a motorcycle for $8,500 with...Ch. 12.III - Prob. 24RECh. 12.III - Prob. 25RECh. 12 - Payment or receipt of equal amounts of money per...Ch. 12 - Prob. 2CRCh. 12 - Prob. 3CRCh. 12 - Prob. 4CRCh. 12 - Prob. 5CRCh. 12 - The table factor for an annuity due is found by...Ch. 12 - 7. Write the formula for calculating the future...Ch. 12 - Prob. 8CRCh. 12 - Prob. 9CRCh. 12 - Prob. 10CRCh. 12 - 11. A(n) ____ fund is an account used to set aside...Ch. 12 - Prob. 12CRCh. 12 - Prob. 13CRCh. 12 - Prob. 14CRCh. 12 - Prob. 1ATCh. 12 - Prob. 2ATCh. 12 - Prob. 3ATCh. 12 - Prob. 4ATCh. 12 - Prob. 5ATCh. 12 - Prob. 6ATCh. 12 - Prob. 7ATCh. 12 - Use Table 12-2 to calculate the present value of...Ch. 12 - Prob. 9ATCh. 12 - Use Table 12-1 to calculate the amount of the...Ch. 12 - Prob. 11ATCh. 12 - Prob. 12ATCh. 12 - Prob. 13ATCh. 12 - Prob. 14ATCh. 12 - Prob. 15ATCh. 12 - Prob. 16ATCh. 12 - Solve the following exercises by using tables.
17....Ch. 12 - Prob. 18ATCh. 12 - Prob. 19ATCh. 12 - Solve the following exercises by using tables....Ch. 12 - Solve the following exercises by using formulas....Ch. 12 - Prob. 22ATCh. 12 - Prob. 23ATCh. 12 - Prob. 24ATCh. 12 - Prob. 25ATCh. 12 - Prob. 26ATCh. 12 - Prob. 27ATCh. 12 - Prob. 28ATCh. 12 - Prob. 29ATCh. 12 - Prob. 30ATCh. 12 - Prob. 31ATCh. 12 - Prob. 32ATCh. 12 - Prob. 33ATCh. 12 - Prob. 34ATCh. 12 - Prob. 35AT
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