Real and Complex Analysis (Rudin)

CHAPTER
THIRTEEN
APPROXIMATIONS BY RATIONAL
FUNCTIONS
Preparation
13.1 Tbe Riemann Spbere It is often convenient in the study of holomorphic
functions to compactify the complex plane by the adjunction of a new point
called 00. The resulting set S2 (the Riemann sphere, the union of R2 and {oo}) is
topologized in the following manner. For any r > 0, let D'( 00; r) be the set of all
complex numbers z such that I z I > r, put D( 00; r) = D'( 00; r) U {oo },. and
declare a subset of S2 to be open if and only if it is the union of discs D(a; r),
where the a's are arbitrary points of S2 and the r's are arbitrary positive numbers.
On S2 - { oo}, this gives of course the ordinary topology of the plane. It is easy to
see that S2 is homeomorphic to a sphere (hence the notation). In fact, a homeomorphism
cp of S2 onto the unit sphere in R3 can be explicitly exhibited: Put'
cp( 00) = (0, 0, 1), and put
(reilJ\ = (2r cos () 2r sin () r2 - 1)
cp J r2 + 1 ' r2 + 1 'r2 + 1
for all complex numbers rei/J. We leave it to the reader to construct the geometric
picture that goes with (1).
Iffis holomorphic in D'( 00; r), we say thatfhas an isolated singularity at 00.
The nature of this singularity is the same as that which the function/, defined in
D'(O; l/r) by /(z) = f(l/z), has at O.
Thus if f is bounded in D'( 00; r), then lim z .... co f(z) exists and is a complex
number (as we see if we apply Theorem 10.20 to J), we define f( 00) to be this
limit, and we thus obtain a function in D( 00; r) which we call holomorphic: note
that this is defined in terms of the behavior of J near 0, and not in terms of
differentiability off at 00.
(1)