UNIVERSITY OF DUBLIN
XMA
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Mathematics
SF Theoretical Physics
JS TSM
Trinity Term 2012
Module MA2325 - Complex Analysis
Saturday, May 19
REGENT HOUSE
9.30 — 11.30
Dr. D. Kitson
Credit will be given for the best FOUR questions. All questions are weighted equally.
Formulae and Tables are available from the invigilators, if required.
Non-programmable calculators are permitted for this examination,—please indicate the make
and model of your calculator on each answer book used.
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1. (a) (5 marks) Solve the equation z 4 = 4i.
(b) (5 marks) Compute the following.
i. sin (π + 2i)
√
ii. Log 1+i
2
(c) (5 marks) Show that the functions z 7→ Re(z), z 7→ z̄ and z 7→ |z| are continuous
on C.
(d) (5 marks) Give an example of a subset of C which is open and star-shaped but not
convex. (Explain your answer).
2. (a) (3 marks) What does it mean to say a function f (z) is complex differentiable at a
point z0 ∈ C?
(b) (3 marks) What are the Cauchy-Riemann equations?
(c) (6 marks) Prove that if a function f = u + iv is complex differentiable at z0 =
x0 + iy0 then u and v are real differentiable at (x0 , y0 ) and the Cauchy-Riemann
equations hold at (x0 , y0 ).
(d) (8 marks) At which points are the following functions complex differentiable?
i. f : C → C,
z 7→ z 2 + z z̄ + z̄ 2
ii. g : C → C,
x + iy 7→ (x3 − 3xy 2 + 4) + i(3x2 y − y 3 − 1)
3. (a) (8 marks) Explain what it means to say a power series
P∞
k=0
ak z k is
i. convergent at a point,
ii. absolutely convergent at a point,
iii. uniformly convergent on a set.
(b) (6 marks) State and prove the Weierstrass M -test for a uniformly convergent series.
(c) (6 marks) Compute the radius of convergence for the following power series.
∞
X
(−1)k k 3
(z − 1)k
k!
k=0
∞
X
k=0
z2
k
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4. (a) (8 marks) Compute the following integral
Z
Im(z) dz
T
where T is the triangular path composed of the line segments [0; 1], [1; 1 + i] and
[1 + i; 0].
(b) (12 marks) State and prove Cauchy’s Theorem for a triangle.
5. (a) (4 marks) State the Residue Theorem and briefly explain any terminology.
(b) (8 marks) Compute the following integral and explain your methods.
Z
ez
dz
2
|z|=2 z (z − 1)(z − 3)
(c) (8 marks) Apply the theory of residues to compute the following improper integral.
Z ∞
1
dx
2
−∞ x + 1
c UNIVERSITY OF DUBLIN 2012