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 f with hook be analytic inside and on a simple closed contour C and let z sub 0 be any point inside C. Then:

f with hook of open paren z sub 0 close paren equals 1 over 2 pi i the contour integral over C of the fraction with numerator f with hook of z and denominator z minus z sub 0 d z

where the integral is taken counterclockwise around C. The factor the fraction with numerator 1 and denominator z minus z sub 0 is called the Cauchy kernel.

The remarkable aspect is that the integral depends only on the values of f with hook on the boundary C yet it recovers f with hook of open paren z sub 0 close paren 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 of z equals the fraction with numerator f with hook of z minus f with hook of open paren z sub 0 close paren and denominator z minus z sub 0 for z is not equal to z sub 0 Since f with hook is analytic, the singularity at z sub 0 is removable, and we can define g of open paren z sub 0 close paren equals f with hook of open paren z sub 0 close paren prime Then g is analytic everywhere inside C

By Cauchy-Goursat:

the contour integral over C of g of z d z equals the contour integral over C of the fraction with numerator f with hook of z minus f with hook of open paren z sub 0 close paren and denominator z minus z sub 0 d z equals 0

Separating the integrals:

the contour integral over C of the fraction with numerator f with hook of z and denominator z minus z sub 0 d z equals f with hook of open paren z sub 0 close paren the contour integral over C of the fraction with numerator 1 and denominator z minus z sub 0 d z

The integral the contour integral over C of the fraction with numerator 1 and denominator z minus z sub 0 d z equals 2 pi i (this can be computed directly for a circle centered at z sub 0, which completes the proof).

Formula for Derivatives

Differentiating the Cauchy integral formula with respect to z sub 0 under the integral sign yields formulas for all derivatives:

f with hook of open paren z sub 0 close paren raised to the n power equals the fraction with numerator n factorial and denominator 2 pi i the contour integral over C of the fraction with numerator f with hook of z and denominator open paren z minus z sub 0 close paren raised to the n plus 1 power 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 equals 1

f with hook prime times open paren z sub 0 close paren equals 1 over 2 pi i the contour integral over C of the fraction with numerator f with hook of z and denominator open paren z minus z sub 0 close paren squared d z

Cauchy's Inequality

Taking C to be a circle of radius R centered at z sub 0 and letting M equals max over the absolute value of z minus z sub 0 equals R of f with hook of z the derivative formula yields:

the absolute value of f with hook of open paren z sub 0 close paren raised to the n power is less than or equal to the fraction with numerator n factorial times M and denominator R to the n-th power

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 the absolute value of f with hook of z is less than or equal to M for all z then for any z sub 0 and any R is greater than 0

the absolute value of f with hook prime times open paren z sub 0 close paren is less than or equal to M over R

Since this holds for arbitrarily large R we must have f with hook prime times open paren z sub 0 close paren equals 0 everywhere. Thus f with hook is constant.

Liouville's theorem has a famous application: it provides a proof of the Fundamental Theorem of Algebra. If p of z is a non-constant polynomial with no roots, then 1 divided by p of 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:

the fraction with numerator 1 and denominator z minus z sub 0 equals the fraction with numerator 1 and denominator open paren z minus a close paren minus open paren z sub 0 minus a close paren equals the fraction with numerator 1 and denominator z minus a times the fraction with numerator 1 and denominator 1 minus the fraction with numerator z sub 0 minus a and denominator z minus a

For the absolute value of z sub 0 minus a is less than the absolute value of z minus a this expands as a geometric series, which when substituted into the Cauchy formula and integrated term by term, yields:

f with hook of open paren z sub 0 close paren equals the sum from n equals 0 to infinity of open paren a sub n close paren times open paren z sub 0 minus a close paren to the n-th power where a sub n equals the fraction with numerator f with hook of a raised to the n power and denominator n factorial

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 of z equals the fraction with numerator f with hook of z and denominator z minus z sub 0 the residue at z sub 0 is f with hook of open paren z sub 0 close paren and the residue theorem gives the contour integral of g of z d z equals 2 pi i times f with hook of open paren z sub 0 close paren

The mean value property follows directly: f with hook of open paren z sub 0 close paren equals the average of f with hook on any circle centered at z sub 0

f with hook of open paren z sub 0 close paren equals 1 over 2 pi the integral from 0 to 2 pi of f with hook of open paren z sub 0 plus r e raised to the i theta power close paren d theta

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 f with hook of open paren z sub 0 close paren equals 1 over 2 pi i the contour integral over C of the fraction with numerator f with hook of z and denominator z minus z sub 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 f with hook of open paren z sub 0 close paren raised to the n power equals the fraction with numerator n factorial and denominator 2 pi i the contour integral over C of the fraction with numerator f with hook of z and denominator open paren z minus z sub 0 close paren raised to the n plus 1 power d z proves that analytic functions are infinitely differentiable. Cauchy's inequality the absolute value of f with hook of open paren z sub 0 close paren raised to the n power is less than or equal to n factorial M divided by R to the n-th power 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.