Question

Please include step-by-step solution.

(iv) Let a be any nonzero complex number. Show that for 12 – 20/ < |al, Z-Zo 2-20 n=0 n = 0 159)..ila). by a dr =0, 6, 11 d =0 Conclude that Z-ZO Z-ZO a for any closed (piecewise) regular curve y that lies in the disk (z – zo] < |al.

Answer 1

RE Page 1 given ato, ats and 12-20 cal Simu 17 series hence she power geometric +2 +2+... - - 1 +2 when Izi<1 I-2 Converges z -z nao I es handy gran and again |- (220) Ja

jer da Now o that to prove a 1-2-zo care where that Y lies is a any closed (piecewise) regular in the disk 12-20l<lal Now to apply Cauchy is the prem. 1 must be analytic or in frer- 1-2-20 0= {z: 12-20lcas. linu f 17) can be expressed as a power series that converges o in D. in an D. analytic function [(z) is analyfie Canchy's theorem hence for so dr = o $17) - Z- I faldzzol 1 dz= 0 Linnlarly a + 2-zodza TY