Question 2 (Unit B2) - 25 marks (a) Evaluate the integral e Jozz + 1) when C is each of the following circles. (i) C = {2 : 12 - 3) = 2} (ii) C = (2 : |2+2-1 = 2} (iii) C = {e : = = 2} (b) Use Lionville's Theorem to prove that the function f(x + iy) = cos(ry) + i sin(ry) is not an entire function. (c) Let p be a non-constant polynomial function and let p' be the derivative of p. By applying the Fundamental Theorem of Algebra to the function q(=) = p(=) - p'(=), prove that there is a complex number a such that p(a) = p'(). Determine whether or not this result remains true if p is replaced by a non-constant entire function? (d) Evaluate the integral cosh z Jo (2 + in/2)4 dz, where C is the circle {2 : (=| = 4}, giving your answer in Cartesian form