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LANCASTER UNIVERSITY
2012 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 complex hyperbolic function cosh, given by cosh z = 12 (ez^ + e−z^ ). (i) Prove that cosh 3z = 4 cosh^3 z − 3 cosh z for all z ∈ C. [4] (ii) If z has Cartesian form x + iy show that cosh z has Cartesian form
cosh x cos y + i sinh x sin y. (^) [3]
(iii) Find all the solutions z ∈ C of the equation cosh z = 0. [3] (iv) Find the singularities of sech z = 1/ cosh z, and determine the radius of convergence of the Maclaurin series for sech. [3]
A2. Consider the power series (^) ∞ ∑ n=
2 n n(n + 1) z
n.
(i) Find the radius of convergence R of this power series. [4] (ii) Prove that this power series converges at all points on the circle C(0, R). [4] (iii) Let f be the function defined by the above power series on the disc D(0, R). Find an expression for f ′(z). [2]
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SECTION A continued
A3. Consider the function f given by f (z) = e
πz z^2 + 2z + 5. (i) Find the poles, their orders and the associated residues of f. [7] (ii) Let γ = C(0, 3) and Γ = C(2i, 3) be circles with radius 3 and centres 0 and 2i. Calculate the integrals (^) ∫
γ
f (z) dz and
Γ
f (z) dz using Cauchy’s Residue Theorem and the aid of a clear diagram. [5]
A4. (a) State Cauchy’s Integral Formula for derivatives for a circular contour. [4] (b) Calculate (^) ∫
C
z − 1 (z^2 + 1)(z + 1)^2 dz, where C is the circular contour C(− 2 , 2). [5]
A5. (a) State the Cauchy–Riemann equations.^ [2] (b) Show that the function given by f (z) = z^2 + z is not holomorphic at any point in C. [4]
SECTION B
B1. (i) Establish the identity 1 2 i − cos θ =^ −^
2 z z^2 − 4 iz + 1 for z = eiθ, where θ is real. [4] (ii) Deduce from the Cauchy Integral Formula that ∫ (^2) π 0
dθ 2 i − cos θ =^ −^
(^2) √πi 5
. [12]
(iii) By considering the imaginary part of the integral in part (ii) calculate ∫ (^2) π 0
dθ 4 + cos^2 θ.^ [4]
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SECTION B continued
B4. (^) (a) Let ϕ(z) =^2 zz ++ 2 i (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 = Re z − 3 } into the circle C(3, 1). [4] (b) Show that the function u defined by
u(x, y) = 2xy − ex^ cos y
is harmonic. Find a harmonic conjugate for u and a holomorphic function f such that u = Re f. [8]
end of exam
