OO Z n ii If exp z E v show that nO expa b exp a exp b iii If sin z 1 show that

Oo z n ii if exp z e v show that no expa b exp a exp

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OO Z n (ii) If exp z = E ~.v show that n=O exp(a + b) = exp a. exp b, (iii) If sin z = 1, show that z is a real number. Va, b c C. 3. (i) State and prove Taylor's theorem for a function f differentiable in the open disc D(a, R), centre a, radius R. O0 (ii) Find the Taylor expansion of (1 - z) -1 of the form E c~(z + i) ~, valid in the n=0 disc D(-i, v/2). Would the expansion be valid in the disc D(-i, 2)? (iii) Find the Laurent expansion, about 0, of (z 3 - 1) -1 valid a) for [z[ < 1, b) Iz[ > 1. MATHM211 PLEASE TURN OVER
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. (i) If f is holomorphic and bounded on C, show that f is constant on C. (ii) Show that if p(z) is a non-constant polynomial with coefficients in C then there exists w C C, with p(w) = O. (iii) If q(z) = 7z 7 + 8z 6 + 3z 5 + 4z 4 + 1, show that sup{Iq(z)l;fz ] = 2} > sup{Jq(z)l;Iz I = 1}. . (i) Let f be a continuous function defined in the upper half plane Im z ~> 0, where f(z) ---, 0 uniformly as Izl --~ c¢. If m > 0, show that ~re~mZ f(z)dz ~ as ~ oc 0 R where FR ---- {Re i°, 0 ~< 8 ~< 7r}. (ii) Evaluate jr0 °° sin x 1)dx" x(x 2 + . (i) Let f be holomorphic inside and on a convex contour 3, except for a finite number of poles at al, ..., an inside 3' with residues R1, ..., P~ respectively. that (ii) Prove that n a positive integer. ~ f(z)dz = 27ri(R1 + ... + P~). cos 2n x + sin 2'~ xdx = zr; n Show MATHM211 END OF PAPER 2
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