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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS SUPPLEMENTARY EXAMINATION – JUNE 2007 MT 2807 - COMPLEX ANALYSIS Date & Time: 26/06/2007 / 9:00 - 12:00 Dept. No. Max. : 100 Marks Answer all the questions. Each question carries 20 marks 1. (a). (i). Prove that a function analytic in B(a, R) can be expressed as a power series. (OR) (ii). State and prove the maximum modulus theorem. (5) (b). (i). Obtain the Leibniz formula for differentiating under the integral sign. (ii). If p(z) is a non-constant polynomial then prove that there is a complex number a with p(a) = 0. (10+5) (OR) (iii). If 0 and 1 are two closed rectifiable curves in G and 0 ~ 1 , then prove that f f , for every function f analytic on G. 0 1 (15) 2. (a). (i). Prove that if C (G , ) is normal then for each z in G, the set { f ( z ) : f } has compact closure in and that is equicontinuous at each point of G. (OR) (ii). State and prove Schwarz Lemma. (5) (b). (i). Let a < b. Let G be the infinite strip {x + iy: a< x < b}. Let f : G C be continuous and let f be analytic on G. Let |f (z)| < B for all z in G. Then prove, with usual notation, that log M(x) is a convex function. (15) (OR) (ii). Prove that a differentiable function f on [a, b] is convex if and only if f’ is increasing. (iii).Given > 0, prove with usual notation that there is a > 0 and a compact set K G such that for f and g C(G, ), sup{d(f(z), g(z)): z K } < (f, g) < . State and prove the converse. (7+8) 3. (a). (i).Let X be a set and let f, f1, f2 … be functions from X into C such that f n ( x) f ( x) uniformly for x in X. If there is a constant a such that Re f(x) ≤ a for all x in X then exp f n ( x) exp f ( x) uniformly for x in X. (OR) z2 (ii). Prove that sin z z 1 2 . n n 1 (b). (i). State and prove the Weierstrass factorization theorem. (15) (OR) (ii). Let (X, d) be a compact metric space and let {gn} be a sequence of continuous functions from X to C such that g n ( x) converges absolutely and uniformly n 1 for x in X. Then prove that the infinite product (1 g n ( x)) converges n 1 absolutely and uniformly for x in X. Also prove that there is an integer N such that f(x) = 0 if and only if gn(x) = –1 for some n, 1 ≤ n ≤ N. n! (iii). Define the gamma function. Prove that ( z ) lim nz . n z ( z 1)( z 2)...( z n) (7+8) 4. (a). (i). State and prove the Jenson’s formula. (OR) (ii). State the Mittag Leffler’s theorem. (5) (b). (i). State and prove the Runge’s theorem on rational functions. (OR) (ii). State and prove the Hadamard’s factorization theorem. (15) 5. (a). (i). Prove that any two bases of a period module are connected by a unimodular transformation. (OR) (ii). Prove that a non-constant elliptic function has always poles. (5) (b). (i). Prove that the sum of the residues of an elliptic function at its poles is zero. (15) (OR) (ii). Prove that an elliptic function can have only a finite number of poles and a finite number of zeros in any mesh. (iii). Prove that a non-constant elliptic function must be of order greater than or equal to two. (9+6) ****************