Q2. Let f be an analytic function defined in a domain D. Show that f must be constant if one of the following statements is true for all z E D. (a) f is real valued. (b) f(z) is analytic. (c) f(z) is constant. (d) Ref(z) or Im f(z) are constant functions. (e) argf is constant. (f) Ref(z) = [Im f(z)] Q3. Work out the following: (a) Use the Milne Thomson method to construct an analytic function f(z) = u+ iv when u + v = ex (cos y sin y) + 2(x + y) + 1. (b) Find the zeros, periods and singularities if any of the following functions cos(2zi +12), sin( 2z i + 13) and tan(2z i + 14). (c) Show that sin hz and cos hz are unbounded functions. Q4. Find the value of the integral 1+2 (3x - y + xi)dz along the real axis from z = 0 to z = 1 and then along a line parallel to the imaginary axis from z = 1 to z = 1 + 2i.