Chapter 35

Analytic Functions

This part of the book is on the fundamentals of complex analysis. I will not try to givetheorems in greatest possible generality. My intent is to give a fairly rigorous presentationof those parts of the subject which have the most interesting applications. I think thatsometimes, when one tries to give the greatest generality and precision, the fundamentalideas are obscured. These are often very simple ideas and it is too bad when they are lost.Complex analysis is quite different than real analysis. It is relatively free of pathology andoften has a much more algebraic flavor than real analysis. I am trying to emphasize thesethings, many of which are very important in both pure and applied math.

The fundamental theorems of Chapter 13 are going to be needed here.

35.1 Cauchy Riemann EquationsOf interest are functions f : U → C where U is an open subset of R2 and we consider R2

to equal C where the ordered pair (x,y) is written as x+ iy. It is customary to write ∂Uto denote the boundary of the open set U . This means U \U whenever U is open and it isuseful to think of it as the edge of U . Thus ∂B is a circle if B is an open ball. This will beused whenever convenient.

As noted earlier in Section 2.3, the complex numbers forms a field. That is, it acts justlike the real numbers. There is a multiplication and addition which satisfy the usual prop-erties which we think numbers should satisfy. Recall from calculus the familiar formula

limh→0

f (z+h)− f (z)h

≡ f ′ (z)

When functions of many variables were encountered earlier, it was necessary to presentthis in another way in terms of little o notation or more directly as

lim|v|→0

|f (x+v)−f (x)−Df (x)v||v|

= 0

We had to do it this way because one cannot divide by a vector. However, in the case wherez ∈ C, no such worry is necessary. The familiar calculus formula can be used becauseindeed, you can divide by a nonzero complex number. This leads to the concept of ananalytic function which will be presented in what follows. We will see that these are just

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