Cauchy's Integral Theorem - Proof II

The fundamental result - Cauchy's Integral Theorem - says roughly:

If C is a simple closed path and w = f(z) is analytic inside and on C, then

ó (ç) õ C  f(z)  dz = 0.

There are two common approaches to this result. The first approaches the Cauchy-Riemann equations and Green's Theorem. The second approach uses less assumptions about the regularity of the derivative f^¢ and builds up the proof by first considering C to be a simple closed triangle and then approximating the general simple closed path by a simple closed polygonal path.

Cauchy's Integral Theorem for a Triangle Using Bolzano-Weierstrass

Theorem. Let T be a simple closed triangle enclosing a region D. Suppose that on D ÈT, w = f(z) is analytic. (No assumption is made on the continuity of f^¢(z).) Then

ó (ç) õ T  f(z)  dz = 0.

Proof: The proof is by contradiction. Suppose there is a triangle T₀ for which

ê ê ó (ç) õ T0  f(z)  dz ê ê = d > 0.

Let the perimeter of T₀ be p₀. Using the midpoints of eachside of T₀, divide T₀ into four congruent triangles, each of perimeter p₁ = [(p₀)/(2¹)].

On at least one of the four triangles, call it T₁,

ê ê ó (ç) õ T1  f(z)  dz ê ê ³ d 41 .

Next, using the midpoints of eachside of T₁, divide T₁ into four congruent triangles, each of perimeter p₂ = [(p₀)/(2²)]. On at least one of the four triangles, call it T₂,

ê ê ó (ç) õ T2  f(z)  dz ê ê ³ d 42 .

Continuing in this manner, we construct a shrinking sequence of similar triangles, T_n, of perimeter p_n = [(p₀)/(2ⁿ)]. with

ê ê ó (ç) õ Tn  f(z)  dz ê ê ³ d 4n .

By the Bolzano-Weierstrass Property, there is a number z₀ within all the triangles. Since f(z) is analytic at z = z₀, near z₀,

f(z) = f(z0) + f¢(z0)·(z - z0) + o(z-z0).

Thus

ó (ç) õ Tn  f(z)  dz = ó (ç) õ Tn  f(z0) + f¢(z0)·(z - z0) + o(z-z0)  dz = 0 + 0 + ó (ç) õ Tn  o(z-z0)  dz = o æ è 1 2n ö ø p0 2n = o æ è 1 4n ö ø .

But [(d)/(4ⁿ)] is not o([1/(4ⁿ)]).

If P is a simple closed polygon, it follows that if f(z) is analytic on and inside P,

ó (ç) õ P  f(z)  dz = 0.

The case of C being a simple closed oriented piecewise C¹ path follows by approximating C by simple closed polygons.

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File translated from T_EX by T_TH, version 3.80.
On 02 Oct 2008, 11:59.