Question

Solve for the three cases n>=0 using Cuachy , n=-1, n<=-2 (cant use cauchy for this one)

4.15. Taking the previous two exercises one step further, fix zo E C and let y be a simple, closed, positively oriented, piecewise smooth path such that zo is inside y. Show that, for any integer n, Tin a - Zo)" dz = S2ni if n= -1, otherwise.

Answer 1

Sal let Zof chrbe simple closed positively oriented, precewise 1 Smooth puth such that zo is inside 2 o it nyo [cz-zol" dz zo sonce. (Z-Zo) "Ifiz)' analytic inside for on 2 i By lawehy them. zzo (7-20) "dz=0 Jo myosot-mann N m © if nog 3m sost-maan Sca-ti de zing more $42-300dt - og det fram i 20 ,- were frzl=1 Then f (tol zo tmy 2 +fm770) = 1 p for mal 64|(23) de femmes et ma m - 1 --Zo) dzs 22 my2 n=1 na-2