LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II (Second Year)
MATHEMATICS & STATISTICS 2 hours
Math 215: Complex Analysis
You should answer ALL Section A questions and THREE Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can
gain there is capped at 40.
SECTION A
A1. Consider the power series
∞
X
n=1
zn
n(n+ 1).
(i) Explain carefully what is meant by the term radius of convergence. [3]
(ii) Find the radius of convergence Rof this power series. [5]
(iii) Prove that this power series converges at all points on the circle |z|=R. [4]
(iv) Let fbe the function defined by the above power series. Find a power series expression
for f0(z). What is the radius of convergence of the series for f0(z)? [3]
A2. (i) Let wbe a non-zero complex number and let [0, w] denote the line segment from 0 to
w. Show that
Z[0,w]
z dz =|w|2
2.[5]
(ii) Prove that the function given by f(z) = zis not complex differentiable at z= 0. [5]
A3. (i) State Liouville’s Theorem, explaining the precise meaning of the hypotheses. [5]
(ii) Explain why the function f(z) = e−z4does not provide a counter-example to Liouville’s
Theorem. [5]
please turn over
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