1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

254 CHAPTER 7 • TAYLOR AND LAURENT SERIES

00
i. Corollary 7.2 If the series L: c,.. (z - a)" converges uniformly to f (z) on the
n ; O
set T and C is a contour contained in T , then

E 1c,,(z-a)"dz=1fc,..(z- a)"dz=1 f(z)dz.

n ; O C Cn;O C

00

• EXAMPLE 7.2 Show that - Log(l -z) = 2:: ~zn, for all z E D1 (0).
  n .=l
  Solution For zo E D 1 (O), we chooser and R so that 0 ::; lzol < r < R < 1,
  thus ensuring that z 0 E Dr (0) and that Dr (0) c DR (0). By Corollary 7.1, the
  00
  geometric series 2:: z" converges uniformly to 1 ~,.on Dr (0). If C is any contour
  n ; O
  contained in Dr (0), Corollary 7.2 gives

l i ~ z dz= f 1 zndz.

C n ; O C

(7-4}

Clearly, the function f (z) = 1 ~, is analytic in the simply connected domain

DR (0), and F (z) = - Log(l - z) is an antiderivative off (z) for all z E DR (0),

where Log is the principal branch of the logarithm. Likewise, g (z) = zn is

analytic in the simply connected domain DR (0), and G (z) = n~l zn+l is an

antiderivative of g (z) for all z E DR (0). Hence, if C is the straight-line segment

joining 0 to zo, we can apply Theorem 6.9 to Equation (7-4) to get

-Log(l -z)l~o = f (-1-zn+l) 'zo'

n ; O n + l o