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LANCASTER UNIVERSITY
2007 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. (a) State de Moivre’s Theorem. [2] (b) Find all solutions of the equation z^3 = − 1 , giving your answers in Cartesian form and expressed in terms of roots (not decimals). Plot the solutions on an Argand diagram. [5] (c) Consider the complex hyperbolic function cosh, given by cosh z = 12 (ez^ + e−z^ ). (i) If z has Cartesian form x + iy show that cosh z has Cartesian form cosh x cos y + i sinh x sin y. (^) [3] (ii) Find the zeros of cosh. [3]
A2. (a) Let Ω ⊂ C and let f be a function Ω → C. What does it mean to say that f is continuous? [2] (b) Let h : C → C be the function defined by h(z) = ¯z. (i) Show that h is continuous. [3] (ii) Show that h is not differentiable at 0. [4] (iii) Is h differentiable anywhere? (Justify your answer.) [3]
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SECTION A continued
A3. Let φ(z) =^2 zz + 2^ −^ ii (z 6 = − 2 i). (i) Express φ as a composition of elementary functions. [6] (ii) Show that φ maps the line {z : Im z = − 34 }^ into the circle C(0, 2). [4]
A4. (i) State the Cauchy-Riemann equations. [2] (ii) Define functions u and v by u(x, y) = (x cos 2x + y sin 2x) e^2 y^ and v(x, y) = (y cos 2x − x sin 2x) e^2 y, and set f (z) = u(x, y)+iv(x, y) for all z = x+iy. Show that f is a holomorphic function. [5]
A5. Show that the function u defined by u(x, y) = x^3 − 3 xy^2 − 3 x is harmonic. Find a harmonic conjugate for u and a holomorphic function f such that u = Re f. [8]
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SECTION B continued
B4. Let f (z) = e
iz z^2 + 2z + 2. (i) Find the poles of f. Calculate the residue of the pole lying in the upper half-plane. [4] (ii) Show that, when |z| = R and Im z ≥ 0, |f (z)| ≤ (^) R (^2) − 21 R − 2. (^) [3]
(iii) By integrating f around a semicircular contour in the upper half-plane, find ∫ (^) ∞ −∞
eix x^2 + 2x + 2 dx.^ [10] (iv) Deduce that (^) ∫ (^) ∞ −∞
sin x x^2 + 2x + 2 dx^ =^ −^
π e sin 1.^ [3]
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