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(math304)[2007](s)final~PPSpider^_10499.pdf
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MATH304, Spring 2007
Final Examination
Time allowed: 100 minutues Course instructor: Prof. Y.K. Kwok
[points]
1. (a) Find the radius of convergence of the following power series
3+z + 3z 2 + z 3 + 3z 4 + z 5 + 3z 6 +
Hint: Use the root test. [2]
(b) Use Weierstrass M-test to prove that
. zn
n=1 n2(n + 1)
is uniformly convergent for |z|1. [2]
(c) Show that
. 4
z
=1+z 4 for |1+z 4|> 1.
(1+z4)n.1
n=1
Furthermore, show that the series is uniformly convergent in the region: [3] |1+z 4|r, where r> 1.
n
. 4
z
Hint: De.ne Sn(z)= , then
(1+z4)k.1
k=1
. 1 .
|Sn(z).(1+z 4)|= ...
. 4)n.1.
(1+z
2. Consider the function
z
f(z)= , =0 and ||.1.
(z .)2
(a) Find the Laurent seriesof thefunctionexpanded at z0 = validin |z.|> 0.
[1]
(b)
Find the Laurent series of the function expanded at z0 =0 valid in
(i) |z|< || [2]
(ii) ||< |z|< . [2] Explain why the Laurent series in |z|< ||reduces to a Taylor series. [1]
(c)
Isit meaningfultode.ne(i) Res(f,),(ii)Res(f,0)? If yes, .nd the value of the residue. If not, explain why? [2]
(d)
Find all possible values of the following integral
z dz, ||.=1.
(z .)2
|z|=1
Distinguish the cases where ||> 1 and ||< 1. [2] 1
3. Let p(z)and q(z)be analytic atz0, while z0 is a simple zero of p(z)and z0 is a triple zero of q(z). That is,
p(z0)=0,p .0
(z0)= (z0) (z0)
q(z0)= q (z0)= q =0,q .0.
=
(a)
Show that f(z)= p(z)/q(z)has a double pole at z0. [1]
(b)
Compute [3] Res(f,z0).
Hint: In terms of Taylor series expansion of p(z)and q(z)at z0, we have
p(z) p (z0)(z .z0)+p (z0)(z .z0)2/2!+
f(z)= = .
(z0)(z .(z0)(z .
q(z) qz0)3/3!+qz0)4/4!+
If f has a double pole at z0, then
Res(f,z0)= lim d [(z .z0)2f].
zz0 dz
4. (a) Locate each of the isolated singularities of the function f(z)= cot , and
z
determine whether it is a removable singularity, a pole or an essential singu-larity. If thesingularity isremovable,thengivethelimit of thefunction atthe point. If thesingularity isapole,thengivetheorderof thepole, and compute the residue at the singularity. [3]
(b) Is z = 0 an isolated singularity of f(z)? Give detailed explanation to your answer. [1]
5. Find the Cauchy principal value of the following improper integral:
.
1
J = dx, 0 < < 1.
0 x(x .4)
[7]
Hint: Choose the branch of the power function z as
logz(lnr+i)i
z = e = e ,z = re ,0 < < 2,
where the branch cut of log z is taken to be along the positive real axis. Note that z = 4 is a simple pole of the integrand function which lies along the positive real axis. Choose the closed contour C as shown:
2
Giveyourjusti.cation thatthe contourintegrals along C and . tend to zero as and . 0, respectively.
Distribution of points in this problem
(i)
Relate the contour integral alo