1. Here's an easy and different derivation of Theorem 2.1 (the residue theorem) using the Cauchy integral formula. Let f be holomorphic on an open set U containing a circle C and its interior, except at a single point 20 in the interior of the circle at which f has a pole. Let n be the order of the pole at Zn and write, for z a 20, f(Z) = (Z 20)_\"9(Z), for some function 9(2) holomorphic on all of U with 9(20) 9E 0. Lf(z)dz= [0 %dz. Interpret the right hand side using the Cauchy integral formula (applied to g). (b) Show that the residue of f at 2:, satises (a) Write the integral reszzzU f(z) : (n i 1)!g(n71)(20) and use this, together with (a), to conclude that f[2)dz : 2m' reszzz0 f(z), C