(a) Evaluate the integral ez z(z+1) dz when C is each of the following circles. (i) C= {z: 2-3|=2} (ii) C= {z:/z+2- i| = 2} (iii) C= {z:|2|= 2} (b) Use Liouville's Theorem to prove that the function f(x+iy) = cos(xy) +isin (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(z)=p(z)-p'(z), prove that there is a complex number a such that p(a) = p'(a). Determine whether or not this result remains true if p is replaced by a non-constant entire function? (d) Evaluate the integral Jo dz, cosh z (z+in/2)4 where C is the circle {z: |z| = 4), giving your answer in Cartesian form. [3] [4] [6] [4] [4]