Complex Algebra and Analysis (1) Show that the function f(z) =zz|, where z=x + jy, is not analytic anywhere in the complex plane. (2) It is given that = = =42-3 and f(z=1+1)=-31. Find f(z) from the above information and the harmonic functions u(x, y) and v(x, y) such that f(z) =u(, y) + Ju(x, y). (3) Let u(x, y)=xy-r+y. Show that u(x, y) is a harmonic function and find its complex conjugate function v(I, y). (4) Show that the integral +00 dx x4 +1 2V/2 using Cauchy's residue theorem. (5) Fund the Laurent series for 2 2 f (z) = (z - 1)(2 - 3) with the region 0