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LANCASTER UNIVERSITY
2011 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. Note: throughout this paper C(z, r) denotes the circle with centre z ∈ C and radius r > 0, described once in the positive sense. SECTION A
A1. Consider the power series (^) ∞ ∑ n=
(− √1)n n z
n (^).
(a) Find the radius of convergence R of this power series. [4] (b) Decide whether this power series converges or diverges at the points z = R and z = −R. [4] (c) Let f : D(0, R) → C be the function defined by the above power series. Find an expres- sion for the derivative f ′. [2] A2. Consider the function f given by f (z) = exp(z (^2) − iz + 2) z(z − 2 i). (a) Find the poles, their orders and the associated residues of f. (Decimal values are not required.) [8] (b) Using Cauchy’s Residue Theorem, calculate the integrals ∫ C(0,1)^ f^ (z)^ dz^ and
C(0,3)^ f^ (z)^ dz with the aid of a clear diagram. (As in (a), no decimal values are required.) [6] please turn over
SECTION A continued
A3. (a) Explain what is meant by saying that a subset Ω of C is starlike. [2] (b) For r > 0, let S(r) = {z ∈ C : | Re z| < r and | Im z| < r}^ and S′(r) = S(r) \ { 0 }. Sketch the set S(r), and prove that S(r) is starlike, whereas S′(r) is not starlike. [8] A4. (a) State Liouville’s Theorem. [3] (b) Explain why the functions f and g given by f (z) = sin(z^2 ) and g(z) = sin(|z|^2 )^ for each z ∈ C do not contradict Liouville’s Theorem. [8]
A5. For each of the following five statements, decide whether it is true or false. No justification of your answers is required. (a) The function f given by f (z) = z/z has a limit as z → 0. (b) The principal branch of the logarithm is defined by log(reiθ) = log r + iθ for r > 0 and θ ∈ (π, 3 π). (c) The principal branch of the logarithm is an entire function. (d) The derivative f ′^ of an entire function f is entire. (e) There exists a quadratic complex polynomial without roots. [5]
please turn over
SECTION B continued
B4. (a) Let^ u^ :^ R^2 →^ R^ be a harmonic function. (i) Using the Cauchy–Riemann equations, prove that the function g : C → C given by g(x + iy) = ∂u ∂x (x, y) − i ∂u ∂y (x, y) , x, y ∈ R , is entire. [4] (ii) Show that there exists a harmonic function v : R^2 → R (the harmonic conjugate of u) such that the function f : C → C given by f (x + iy) = u(x, y) + iv(x, y) is entire. You may use general results provided that you give clear reference to them. [6] (b) (i) Determine constants α, β ∈ R such that the function u : R^2 → R given by u(x, y) = − 2 x^3 + αx^2 y + βxy^2 + y^3 , x, y ∈ R , is harmonic. [5] (ii) For such constants, find a harmonic conjugate of u. [5]
end of exam
