Cauchy Integral Formula for Disks

Cauchy Integral Formula

We are going to use the homotopy invariance theorem to prove a fundamental result in the study of holomorphic functions, the Cauchy integral formula, which relates the value of a holomorphic function on the boundary of a domain to the value in the interior. Note that generally a function is not defined on the boundary of its domain, so to talk about boundary values of holomorphic functions, we will need our functions to be defined on a slightly bigger set:

Definition

If is a domain, then we write for

e.g. but .

Cauchy Integral Formula for Disks

Let . Then for ,

Proof Let . The path is CP-homotopic to via Hence for all , by the homotopy invariance theorem ,we have We will show that , and since actually does not depend on , we will get the desired result. Observe that , hence we have Since is continuous, this goes to as , so we conclude that .

Remark

The Cauchy integral formula tells us that holomorphic functions are very special: the values in the inside of the disk are determined by the values on the boundary. This is very false if you’re only talking about continuous functions. The single variable analog would be to say that a differentiable function was determined by its values at a and b, which is ridiculous in real analysis.

e.g. For , we have Now if is a real number, we can finally get

Complex techniques for real integrals

This is a fact entirely about real integrals, which does not appear very friendly to real analysis methods. Though this example is somewhat contrived, it demonstrates that complex techniques can be powerful tools for solving problems that, on first glance, have nothing to do with complex analysis. We’ll see more examples and techniques later for evaluating real integrals using complex methods, and these techniques are used heavily in large parts of mathematics and physics.

Derivatives of Holomorphic Functions

We’ll now use the integral formula to show that if , then .

Lemma

Let be an integer, then there exists a polynomial such that for all with , we have

Proof We’ll put the right-hand side over a common denominator and all of the terms with powers and cancel, leaving just terms with or higher. By binomial theorem, for some polynomial . So we have Hence the lemma is proved, with .

Proposition

Let . Define by Then is arbitrarily many times -differentiable, and

Proof We prove by induction on . The base case holds automatically by its expression. Assume the result holds for . Now consider the following expression, applying the above lemma with and : When is sufficiently small, the integrand is bounded by some . Therefore by the ML estimate, the above expression goes to as . Hence by principal of mathematical induction, holds for all .

Corollary

Let . Then is infinitely many times -differentiable on and for

Proof By the above proposition and Cauchy integral formula on , is arbitrarily many times -differentiable on .

Corollary

Let be a domain and . Then and so is infinitely many times -differentiable on .

Proof Fix some . Since is open, we can choose some , then . We apply the above corollary to get is infinitely many times -differentiable on , in particular -differentiable at . Since is arbitrary, is infinitely many times -differentiable on .