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LANCASTER UNIVERSITY
2009 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. Let f (z) =
∑^ ∞
n=
(−1)n 2 n − 1 z
n.
(i) Find the radius of convergence of the series for f (z) and determine whether this series converges when z = 1. [3] (ii) Find the series for f ′(z) and calculate its radius of convergence. [4] (iii) Does the series for f ′(z) converge at some point on its circle of convergence? [3]
A2. Give a formula expressing sinh z in terms of the exponential function. Using this or otherwise, find all the solutions z ∈ C of the equation sinh z = 0. [5]
A3. (i) State Cauchy’s Theorem for a triangle. [3] (ii) Let R be a rectangle, and let f be holomorphic on and inside R. Use (i) to find
R f^ (z)^ dz.^ [4]
A4. (i) State Cauchy’s Integral Formula for derivatives for a circular contour. [4] (ii) Calculate (^) ∫
C
sin z (z − 1)^3 dz, where C is the circular contour C(0, 2). [5]
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SECTION A continued
A5. (^) (a) State Liouville’s Theorem, explaining the precise meaning of the hypotheses. [5] (b) Give (without proof) an example of an entire function that is non-constant and that has no zeros. [2]
A6. Let ϕ(z) = i i^ −+^ zz (z = −i). (i) Express ϕ as a composition of elementary functions. [5] (ii) Find the M¨obius transformation that is the inverse of ϕ. [3] (iii) Show that ϕ maps the line {z : Im z = − 2 } into the circle C(− 2 , 1). [4]
SECTION B
B1. (^) (a) (i) Suppose z = 1 and N ∈ N. Show that
1 + z + z^2 + · · · + zN^ =^1 −^ z
N + 1 − z.^ [5] (ii) Show that (^) ∞ ∑ n=
zn^ = (^1) −^1 z.
when |z| < 1. [4] (iii) Let 0 ≤ r < 1 and θ ∈ R. Establish the formula
1 + r cos θ + r^2 cos 2θ + · · · = (^1) −^1 2 −r cosr^ cos θ +θ r 2. (^) [5]
(b) Let f (z) = (^2) z (^2) −^1 5 z + 3. Rewrite the expression for f (z) using partial fractions. Find a series expression for f (z) of the form f (z) = ∑∞ n=0 anzn, and determine where this series converges. [6]
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