Chapter 56

Approximation By Rational Functions56.1 Runge’s Theorem

Consider the function, 1z = f (z) for z defined on Ω ≡ B(0,1) \ {0} = B′ (0,1) . Clearly

f is analytic on Ω. Suppose that you could approximate f uniformly by polynomials onann(0, 1

2 ,34

), a compact subset of Ω. Then, there would exist a suitable polynomial p(z) ,

such that∣∣∣ 1

2πi∫

γf (z)− p(z)dz

∣∣∣< 110 where here γ is a circle of radius 2

3 . However, this is

impossible because 12πi∫

γf (z)dz = 1 while 1

2πi∫

γp(z)dz = 0. This shows you can’t expect

to be able to uniformly approximate analytic functions on compact sets using polynomials.This is just horrible! In real variables, you can approximate any continuous function ona compact set with a polynomial. However, that is just the way it is. It turns out that theability to approximate an analytic function on Ω with polynomials is dependent on Ω beingsimply connected.

All these theorems work for f having values in a complex Banach space. However, Iwill present them in the context of functions which have values inC. The changes necessaryto obtain the extra generality are very minor.

Definition 56.1.1 Approximation will be taken with respect to the following norm.

|| f −g||K,∞ ≡ sup{|| f (z)−g(z)|| : z ∈ K}

56.1.1 Approximation With Rational FunctionsIt turns out you can approximate analytic functions by rational functions, quotients of poly-nomials. The resulting theorem is one of the most profound theorems in complex analysis.The basic idea is simple. The Riemann sums for the Cauchy integral formula are rationalfunctions. The idea used to implement this observation is that if you have a compact subset,K of an open set, Ω there exists a cycle composed of closed oriented curves

{γ j

}n

j=1which

are contained in Ω\K such that for every z ∈ K,∑nk=1 n(γk,z) = 1. One more ingredient is

needed and this is a theorem which lets you keep the approximation but move the poles.To begin with, consider the part about the cycle of closed oriented curves. Recall

Theorem 51.7.25 which is stated for convenience.

Theorem 56.1.2 Let K be a compact subset of an open set Ω. Then there exist continuous,closed, bounded variation oriented curves

{γ j

}m

j=1for which γ∗j ∩K = /0 for each j, γ∗j ⊆

Ω, and for all p ∈ K,m

∑k=1

n(p,γk) = 1.

andm

∑k=1

n(z,γk) = 0

for all z /∈Ω.

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