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(math304)[2006](s)final~PPSpider^_10498.pdf
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MATH304, Spring 2006

Final Examination
Time allowed: 2 hours Course instruction: Prof. Y.K. Kwok
[points]

1. Let R> 0 be the radius of convergence of the series anz n . Show that the radius n=0

of convergence of the related series (Rean)z n is at least R. When does equality
n=0

of the two radii of convergence hold?
Hint: Observe that |an||Rean|and use the root test. [5]

1
2.
Expand f(z) = in a Laurent series valid for |z +1|> 2. Can we

(z +1)(z +3) use this Laurent series to classify the nature of the isolated singularity z = .1 of f(z)?Ifyes,determine whetherz = .1is apole, removable singularity oressential singularity. If not, why not. [5]

3.
A proper rational function with simple poles only can be represented by


b0zn + + bn.1z + bn
f(z)= , n<k,
(z .z1)(z .z2) (z .zk)
where z1,z2, ,zk are all distinct.

(a) Show that the corresponding partial fraction decomposition of f(z)takes the form [3]
c1 c2 ck
f(z)= + + + ,
z .z1 z .z2 z .zk
where cj =Res(f,zj),j =1,2, ,k.

(b) Apply the result to .nd the partial fraction decomposition of [3] 1
f(z)= .
zn .1
th 2ki/n,k
Hint: The nroots of unity are e=0,1, ,n .1.
4. Compute the residue at each of the isolated singularities of cos z
f(z)= .
z2 (z .)3
+fg
Hint:(fg) = fg+2fg . Here, fand fdenote the .rst and second order derivatives of f. [7]
5. Let f be a function which is analytic at the point z0. Show that z0 is a removable singular point of the function
f(z)
g(z)=
z .z0
when f(z0)=0. Showthatwhen f(z0).0, thepoint z0 is a simple pole of g. Find

= Res(g,z0). [4]
6. Consider the contour integral
. 2iz
1.e
dz
z2
C
where C is the closed contour depicted in the following .gure.

The closed contour C consists of the in.nitely large upper semi-circle CR of radius R, the line segments (.R,.) and (,R) along the x-axis, and the in.nitesimal upper semi-circle C of radius . around z =0.
2iz
1.e
(a) State the Jordan Lemma and use it to show that the integration of
z2
along the upper semi-circle CR vanishes as R . Note that [3]
2iz 2iz
1.e1 e
dz = dz . dz.
z2 z2 z2
CR CR CR
(b) Why we need to include the indented semi-circle C at z = 0 in the closed contour? Evaluate [3]
. 2iz
1.e
dz.
z2
C
(c) Find the Cauchy principal value of the improper integral
.
sin2 x
dx.
x2
.
[4]
. .
sin2 x 1.e2iz
Hint: Relate dx to dz and use the identity:
x2 z2
. C
cos2x =1.2sin2 x.
7. Find the bilinear transformation that maps the interior of the circle: |z|< 1 in the 1+i
z-plane to the interior of the circle: |w|< 1 in the w-plane. It maps z1 = to
22 w1 =0 and z2 =0 to w2 =1/2. [6]
Hint: Find the inversion point of z1 with respect to |z|=1 and use the property
that a pair of symmetric points are mapped to a pair of symmetric point
under a bilinear transformation.

8. Consider