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

⌠ (⎜) ⌡ 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 ∪T, w = f(z) is analytic. (No assumption is made on the continuity of f^′(z).) Then

⌠ (⎜) ⌡ T  f(z)  dz = 0.

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

⎢ ⎢ ⌠ (⎜) ⌡ T0  f(z)  dz ⎢ ⎢ = δ > 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,

⎢ ⎢ ⌠ (⎜) ⌡ T1  f(z)  dz ⎢ ⎢ ≥ δ 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,

⎢ ⎢ ⌠ (⎜) ⌡ T2  f(z)  dz ⎢ ⎢ ≥ δ 42 .

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

⎢ ⎢ ⌠ (⎜) ⌡ Tn  f(z)  dz ⎢ ⎢ ≥ δ 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

⌠ (⎜) ⌡ Tn  f(z)  dz = ⌠ (⎜) ⌡ Tn  f(z0) + f′(z0)·(z − z0) + o(z−z0)  dz = 0 + 0 + ⌠ (⎜) ⌡ Tn  o(z−z0)  dz = o ⎛ ⎝ 1 2n ⎞ ⎠ p0 2n = o ⎛ ⎝ 1 4n ⎞ ⎠ .

But [(δ)/(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,

⌠ (⎜) ⌡ 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 4.03.
On 20 Feb 2012, 10:21.