1778 CHAPTER 56. APPROXIMATION BY RATIONAL FUNCTIONS

and the Weierstrass M test can be applied because∣∣∣∣ zzk

∣∣∣∣< 12

on this set. Therefore, by Corollary 56.1.7, Sk (z) , being a polynomial in 1z−zk

, has a powerseries which converges uniformly to Sk (z) on Kk. Therefore, there exists a polynomial,Pk (z) such that

||Pk−Sk||B(0,|zk|/2),∞ <12k .

Let

Q(z)≡∞

∑k=1

(Sk (z)−Pk (z)) . (56.2.12)

Consider z ∈ Km and let N be large enough that if k > N, then |zk|> 2 |z|

Q(z) =N

∑k=1

(Sk (z)−Pk (z))+∞

∑k=N+1

(Sk (z)−Pk (z)) .

On Km, the second sum converges uniformly to a function analytic on int(Km) (interior ofKm) while the first is a rational function having poles at z1, · · · ,zN . Since any compact setis contained in Km for large enough m, this shows Q(z) is meromorphic as claimed and haspoles with the given singularities.

Now consider the case where the poles are at {zk}∞

k=0 with z0 = 0. Everything is similarin this case. Let

Q(z)≡ S0 (z)+∞

∑k=1

(Sk (z)−Pk (z)) .

The series converges uniformly on every compact set because of the assumption that

limn→∞|zn|= ∞

which implies that any compact set is contained in Kk for k large enough. Choose N suchthat z ∈ int(KN) and zn /∈ KN for all n≥ N +1. Then

Q(z) = S0 (z)+N

∑k=1

(Sk (z)−Pk (z))+∞

∑k=N+1

(Sk (z)−Pk (z)) .

The last sum is analytic on int(KN) because each function in the sum is analytic due to thefact that none of its poles are in KN . Also, S0 (z)+∑

Nk=1 (Sk (z)−Pk (z)) is a finite sum of

rational functions so it is a rational function and Pk is a polynomial so zm is a pole of thisfunction with the correct singularity whenever zm ∈ int(KN).

56.2.3 Functions Meromorphic On ĈSometimes it is useful to think of isolated singular points at ∞.