In all problems involving days, a 360 -day year is assumed. When annual rates are requested as an answer, express, express the rate as a percentage, correct to three decimal places, unless directed otherwise. Round dollar amounts to the nearest cent. What is the purchase price of a 50 -day T-bill with a maturity value of $ 1 , 000 that earns an annual interest rate of 5.53 % ?
In all problems involving days, a 360 -day year is assumed. When annual rates are requested as an answer, express, express the rate as a percentage, correct to three decimal places, unless directed otherwise. Round dollar amounts to the nearest cent. What is the purchase price of a 50 -day T-bill with a maturity value of $ 1 , 000 that earns an annual interest rate of 5.53 % ?
In all problems involving days, a
360
-day year is assumed. When annual rates are requested as an answer, express, express the rate as a percentage, correct to three decimal places, unless directed otherwise. Round dollar amounts to the nearest cent.
What is the purchase price of a
50
-day T-bill with a maturity value of
$
1
,
000
that earns an annual interest rate of
5.53
%
?
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)
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