1770 CHAPTER 56. APPROXIMATION BY RATIONAL FUNCTIONS

Lemma 56.1.6 Let ∑∞n=0 an (z) and ∑

∞n=0 bn (z) be two convergent series for z ∈ K which

satisfy the conditions of the Weierstrass M test. Thus there exist positive constants, An andBn such that |an (z)| ≤ An, |bn (z)| ≤ Bn for all z ∈ K and ∑

∞n=0 An < ∞,∑∞

n=0 Bn < ∞. Thendefining the Cauchy product,

cn (z)≡n

∑k−0

an−k (z)bk (z) ,

it follows ∑∞n=0 cn (z) also converges absolutely and uniformly on K because cn (z) satisfies

the conditions of the Weierstrass M test. Therefore,

∞

∑n=0

cn (z) =

(∞

∑k=0

ak (z)

)(∞

∑n=0

bn (z)

). (56.1.3)

Proof:

|cn (z)| ≤n

∑k=0|an−k (z)| |bk (z)| ≤

n

∑k=0

An−kBk.

Also,

∞

∑n=0

n

∑k=0

An−kBk =∞

∑k=0

∞

∑n=k

An−kBk

=∞

∑k=0

Bk

∞

∑n=0

An < ∞.

The claim of 56.1.3 follows from Merten’s theorem. This proves the lemma.

Corollary 56.1.7 Let P be a polynomial and let ∑∞n=0 an (z) converge uniformly and abso-

lutely on K such that the an satisfy the conditions of the Weierstrass M test. Then there existsa series for P(∑∞

n=0 an (z)) ,∑∞n=0 cn (z) , which also converges absolutely and uniformly for

z ∈ K because cn (z) also satisfies the conditions of the Weierstrass M test.

The following picture is descriptive of the following lemma. This lemma says that ifyou have a rational function with one pole off a compact set, then you can approximate onthe compact set with another rational function which has a different pole.

V a

b

K