 # 1. (a) (i) State the Cauchy-Riemann equations. (ii) Verify them f...

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1. (a) (i) State the Cauchy-Riemann equations. (ii) Verify them for the function f : C C defined by 4v/2. (10) (b) (i) State the definition of a bounded set in C. (ii) State the definition of a compact set in C. (10) (c) Find the radius of convergence of the following series: 3 zn 3n + n=0 (10) (d) How many zeros, counted with multiplicity, has the function f(z)=25-24+ = - 32z³ + x2 + (12 + 35i) Z + i on the annulus {z € C : 1(10) (e) Find the Laurent series expansion of the function 1 - which is valid in the annulus where 1 < z < 4. (10)  2. (a) Let U be a domain. Let r be a positively oriented, simple closed contour with its image and interior lying entirely within U. Suppose that a is a point in the interior of r and f is a holomorphic function on U. State Cauchy's Integral Formula for f (a). (5) (b) Is the set D = {z E C Re(z) > 1 or Re(z) < -1}. - open? Justify your answer (10) (c) Let r be the unit circle {x € C : |z| = 1} traversed once anticlockwise. Prove that 3z7 sin 2 dz 67e. 425 - 3 r (10) 4. (a) Find the order of each pole of f and calculate the residue of f at each simple pole. (8) (b) Consider the Laurent expansion of f(z) about Z = 0: 8 f(2)= E 6.2" = n=-N where N is the order of the pole of f at 0. Calculate b_2 and b_1. (8) (c) Evaluate Sr f(z) dz, where r is the circular contour, traversed anticlockwise, with centre 0 and radius 3/2. (9) 

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