MAS332 SCHOOL OF MATHEMATICS AND STATISTICS Complex Analysis Answer four Autumn Semester 2013-2014 2 hours 30 minutes questions. If you answer more than four questions, only your best four will be counted. MAS332 1 Turn Over MAS332 1 (i) Express both of the following in the form x + iy : 13 − i ; 1 − 2i (ii) Express (1 − i)11 . (4 marks) (1 − i)13 √ ( 3 − i)11 in the form reiθ with r > 0 and −π < θ ≤ π . (4 marks) (iii) State, without proof, the triangle inequalities for | z + w| and | z − w| . Show that, if | z| ≤ 1 , then 3z − 4 1 ≤ 7. ≤ 5 2z + 3 (4 marks) (iv) Write down the denitions of cosh z and sinh z . Find all the solutions of the following equation: 2 cosh z + sinh z = i. (v) (5 marks) The path γ is the arc of the circle | z + 1| = 1 from 0 to −2 given by z = −1 + e (0 ≤ t ≤ π). Evaluate it Z Z z¯ dz , γ z 3 cos (z 4 ) dz . (4 marks) γ (vi) Find all the sixth roots of −1. Hence express x6 + 1 as the product of three real quadratic factors. (4 marks) MAS332 2 Continued MAS332 2 (i) function. State, without proof, the Cauchy-Riemann equations for a dierentiable (1 mark) (a) Let g(z) = 4z − 3z for all z ∈ C. Prove that g is nowhere dierentiable. (3 marks) (b) The function h is analytic in the complex plane and Im(h(z)) + Re(h(z)) = 2 for all z ∈ C . Show that h is constant. (5 marks) (ii) In each of the following cases, determine whether there is a function k analytic on C with Re (k(x + iy)) = u(x, y), giving reasons for your answers: (a) (b) u(x, y) = cosh x cosh y , u(x, y) = x3 − 3xy 2 − 2y + 1 . When k exists, nd an explicit expression for k(z) in terms of z and show that you have found all functions the satisfying the conditions. (8 marks) (iii) Let the path α from 1 to −3 , consist of the straight line segment from 1 to 1 + 3i , followed by the straight line segment from 1 + 3i to −3 + 3i , followed by the straight line segment from −3 + 3i to −3 . Sketch α. Use the ML estimate to show that Z MAS332 α ez sin z dz z2 ≤ 10 e cosh 3. 3 (8 marks) Turn Over MAS332 3 State, without proof, Cauchy's Theorem and Cauchy's Integral Formulae for a function and for its derivatives. Your statement should include conditions under which the results are valid. (7 marks) Let γ be the square contour with vertices 2, 2i, −2, −2i described in the anti-clockwise direction. Without using the Residue Theorem, evaluate (i) Z γ (iii) Z γ sin (πz) dz , 3z − 1 (ii) Z γ ez dz , z 2 (z + 3) (iv) Z γ ez + 1 dz , z2 + 9 ez dz . z(z + 1) (14 marks) Let the contour α be the circle |z − 1| = 2 described in the positive direction. Evaluate Z (z 2 + z¯) dz . α (4 marks) MAS332 4 Continued MAS332 4 (i) Let f have a pole of order k at α. Prove that the residue of f at the point α is given by Res {f ; α} = 1 dk−1 lim [(z − α)k f (z)]. k−1 z→α (k − 1)! dz (5 marks) (ii) For each of the following functions, nd all the singularities in C. Classify these singularities giving reasons for your answers and evaluate the residue at each of them: (a) cos (πz) , (z − 1)2 (b) z exp MAS332 (4 marks) ez 1 , z−1 (4 marks) (c) eπz , eπz + 1 (5 marks) (d) 1 + cos (πz) , (z − 1) 2 (3 marks) (e) 1 + cos (πz) . (z − 1) 5 (4 marks) 5 Turn Over MAS332 5 (i) State, without proof, Cauchy's Residue Theorem. Your statement should include conditions under which the result is valid. (4 marks) Let γ be the triangular contour with vertices 2, 2i, −2i described in the anticlockwise direction. Evaluate Z γ sin πz dz , (2z + 1) cos πz (ii) Z (z + 1) cos γ 1 z−1 (11 marks) Prove that Z ∞ −∞ (x2 x sin x dx = + 1)(x2 + 4) π(e − 1) . 3e2 End of Question Paper MAS332 dz . 6 (10 marks)

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