Real and Complex Analysis (Rudin)

APPROXIMATIONS BY RATIONAL FUNCTIONS 269
a grid of horizontal and vertical lines in the plane, such that the distance
between any two adjacent horizontal lines is '1, and likewise for the vertical
lines. Let Q1> ... , Qm be those squares (closed 2-cells) of edge '1 which are
formed by this grid and which intersect K. Then Qr c Q for r = 1, ... , m.
If ar is the center of Qr and ar + b is one of its vertices, let Yrk be the
oriented interval
and define
oQr = Yrl -+ Yr2 -+ Yr3 -+ Yr4 (r = 1, ... , m). (3)
It is then easy to check (for example, as a special case of Theorem 10.37, or
by means of Theorems 10.11 and 10.40) that
(2)
IndaQ• (IX) = {~
if IX is in the interior of Qr'
if IX is not in Qr.
(4)
Let 1: be the collection of all Yrk (1 ~ r ~ m, 1 ~ k ~ 4). It is clear that 1:
is balanced. Remove those members of 1: whose opposites (see Sec. 10.8) also
belong to 1:. Let «I> be the collection of the remaining members of 1:. Then «I>
is balanced. Let r be the cycle constructed from «1>, as in Sec. 13.4.
If an edge E of some Qr intersects K, then the two squares in whose
boundaries E lies intersect K. Hence 1: contains two oriented intervals which
are each other's opposites and whose range is E. These intervals do not occur
in «1>. Thus r is a cycle in Q - K.
The construction of «I> from 1: shows also that
m
Indr (IX) = L Ind aQ • (IX)
r= 1
(5)
if IX is not in the boundary of any Qr. Hence (4) implies
if IX is in the interior of some Qr'
if IX lies in no Qr.
(6)
If Z E K, then z ¢ r*, and z is a limit point of the interior of some Qr.
Since the left side of (6) is constant in each component of the complement of
r*, (6) gives
Indr (z) = {~
if z E K,
ifz¢Q.
(7)
Now (1) follows from Cauchy's theorem 10.35.
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