1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 559

The right-hand side of (8.7-4) can be made as small as we please by
taking n sufficiently large, provided that II -zl/(1 -lzl) remains bounded
inside the unit circle, i.e., as long as

II -zl < k

1-lzl -

(8.7-5)

for some real constant k > 1. Inequality (8.7-5) holds in a region bounded

by the curve

II -zl.= k(l - lzl)
If we choose k > 2, that region contains the sector <1 = ABCD shown
in Fig. 8.4 (shaded portion), where chords AP, AQ form with AO an angle
0 < a <! 1T defined by cos a =:= 2 / k. In fact, for any point z in this
sector we have

1-z = rei/3

where 0 ~ r ~AB= OAcosa = cosa and l,81 ~a, and it follows froni
triangle 0 Az that

lzl^2 = 1 + r^2 - 2r cos ,B ~ 1 + r^2 - 2r cos a ~ 1 +'('cos a - 2r cos a

1

= 1 - r cos a ~ 1 - r cos a + 4" r^2 cos^2 a

so that

lzl ~ 1-%rcosa

and

A

x

Fig. 8.4