Cauchy Integral Formula

Introduction

The Cauchy integral formula is one of the most remarkable results in all of mathematics. It states that for an analytic function, the value at any interior point is completely determined by its values on the boundary. This is radically different from real analysis, where knowing a function on a circle tells us nothing about its values inside.

This formula has profound consequences: it implies that analytic functions are infinitely differentiable, that they equal their Taylor series, and that bounded entire functions must be constant (Liouville's theorem). The Cauchy integral formula is the foundation upon which much of complex analysis is built.

The formula also has practical applications: it provides a powerful method for evaluating integrals, computing derivatives, and understanding the behavior of complex functions. Its elegance lies in expressing local information (the value at a point) through global information (an integral around a curve).

Statement of the Theorem

Let ƒ be analytic inside and on a simple closed contour C and let (z_0) be any point inside C. Then:

ƒ((z_0))=1/(2*π*i)*(∮_C)(ƒ(z)/(z−(z_0))*d(z))

where the integral is taken counterclockwise around C. The factor 1/(z−(z_0)) is called the Cauchy kernel.

The remarkable aspect is that the integral depends only on the values of ƒ on the boundary C yet it recovers ƒ((z_0)) at any interior point. The boundary values completely determine the interior.

Proof Outline

The proof uses the Cauchy-Goursat theorem (that integrals of analytic functions around closed contours vanish) together with a deformation argument.

Consider the function g(z)=(ƒ(z)−ƒ((z_0)))/(z−(z_0)) for z≠(z_0) Since ƒ is analytic, the singularity at (z_0) is removable, and we can define g((z_0))=ƒ((z_0))′ Then g is analytic everywhere inside C

By Cauchy-Goursat:

(∮_C)(g(z)*d(z))=(∮_C)((ƒ(z)−ƒ((z_0)))/(z−(z_0))*d(z))=0

Separating the integrals:

(∮_C)(ƒ(z)/(z−(z_0))*d(z))=ƒ((z_0))*(∮_C)(1/(z−(z_0))*d(z))

The integral (∮_C)(1/(z−(z_0))*d(z))=2*π*i (this can be computed directly for a circle centered at (z_0), which completes the proof).

Formula for Derivatives

Differentiating the Cauchy integral formula with respect to (z_0) under the integral sign yields formulas for all derivatives:

ƒ((z_0))(n)=n!/(2*π*i)*(∮_C)(ƒ(z)/((z−(z_0))(n+1))*d(z))

This extraordinary result shows that analytic functions are infinitely differentiable—all derivatives exist and can be expressed as contour integrals. This is in stark contrast to real analysis, where a function can be differentiable once but not twice.

For n=1

ƒ′*((z_0))=1/(2*π*i)*(∮_C)(ƒ(z)/((z−(z_0))2)*d(z))

Cauchy's Inequality

Taking C to be a circle of radius R centered at (z_0) and letting M=(max_|z−(z_0)|=R)(ƒ(z)) the derivative formula yields:

|ƒ((z_0))(n)|≤(n!⋅M)/(Rn)

This Cauchy inequality bounds derivatives in terms of the maximum of the function. It is fundamental for proving Liouville's theorem and for estimating Taylor coefficients.

Liouville's Theorem

A bounded entire function (analytic on all of C must be constant. This remarkable result follows from Cauchy's inequality.

Proof: If |ƒ(z)|≤M for all z then for any (z_0) and any R>0

|ƒ′*((z_0))|≤M/R

Since this holds for arbitrarily large R we must have ƒ′*((z_0))=0 everywhere. Thus ƒ is constant.

Liouville's theorem has a famous application: it provides a proof of the Fundamental Theorem of Algebra. If p(z) is a non-constant polynomial with no roots, then 1/p(z) would be bounded and entire, hence constant—contradiction.

Connection to Taylor Series

The Cauchy integral formula leads directly to Taylor series. Using the geometric series expansion:

1/(z−(z_0))=1/((z−a)−((z_0)−a))=1/(z−a)⋅1/(1−((z_0)−a)/(z−a))

For |(z_0)−a|<|z−a| this expands as a geometric series, which when substituted into the Cauchy formula and integrated term by term, yields:

ƒ((z_0))=(∑_n=0^∞)((a_n))*((z_0)−a)n where (a_n)=(ƒ(a)(n))/n!

This proves that every analytic function equals its Taylor series within the disk of analyticity—a property unique to complex analysis.

Connection to Other Concepts

The Cauchy integral formula is a special case of the residue theorem. For g(z)=ƒ(z)/(z−(z_0)) the residue at (z_0) is ƒ((z_0)) and the residue theorem gives (∮_)(g(z)*d(z))=2*π*i⋅ƒ((z_0))

The mean value property follows directly: ƒ((z_0)) equals the average of ƒ on any circle centered at (z_0)

ƒ((z_0))=1/(2*π)*(∫_0^2*π)(ƒ*((z_0)+r*e(i*θ))*d(θ))

The maximum modulus principle (an analytic function attains its maximum modulus on the boundary) also follows from the Cauchy formula.

Summary

The Cauchy integral formula ƒ((z_0))=1/(2*π*i)*(∮_C)(ƒ(z)/(z−(z_0))*d(z)) expresses the value of an analytic function at an interior point as a contour integral over the boundary. This encapsulates the remarkable rigidity of analytic functions.

The generalization to derivatives ƒ((z_0))(n)=n!/(2*π*i)*(∮_C)(ƒ(z)/((z−(z_0))(n+1))*d(z)) proves that analytic functions are infinitely differentiable. Cauchy's inequality |ƒ((z_0))(n)|≤n!*M/Rn bounds derivatives.

Consequences include Liouville's theorem (bounded entire functions are constant), the equality of analytic functions with their Taylor series, the mean value property, and the maximum modulus principle. The Cauchy integral formula is the cornerstone of complex analysis.