Transcribed Text
1.
(a) (i) State the CauchyRiemann equations.
(ii) Verify them for the function f : C
C defined by
4v/2.
(10)
(b) (i) State the definition of a bounded set in C.
(ii) State the definition of a compact set in C.
(10)
(c) Find the radius of convergence of the following series:
3
zn
3n +
n=0
(10)
(d) How many zeros, counted with multiplicity, has the function
f(z)=2524+ =  32z³ + x2 + (12 + 35i) Z + i
on the annulus {z € C : 1(10)
(e) Find the Laurent series expansion of the function
1

which is valid in the annulus where 1 < z < 4.
(10)
[50]
2. (a) Let U be a domain. Let r be a positively oriented, simple
closed contour with its image and interior lying entirely within
U. Suppose that a is a point in the interior of r and f is a
holomorphic function on U. State Cauchy's Integral Formula for
f (a).
(5)
(b) Is the set
D = {z E C Re(z) > 1 or Re(z) < 1}. 
open? Justify your answer
(10)
(c) Let r be the unit circle {x € C : z = 1} traversed once
anticlockwise. Prove that
3z7 sin 2
dz
67e.
425  3
r
(10)
4.
(a) Find the order of each pole of f and calculate the residue of f at
each simple pole.
(8)
(b) Consider the Laurent expansion of f(z) about Z = 0:
8
f(2)= E 6.2" =
n=N
where N is the order of the pole of f at 0. Calculate b_2 and b_1.
(8)
(c) Evaluate Sr f(z) dz, where r is the circular contour, traversed
anticlockwise, with centre 0 and radius 3/2.
(9)
[25]
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