The Cauchy Theorem

Abstract

Runge’s Theorem can be used to prove Cauchy’s Theorem. This will require the elements of integration theory described in Chapter 2. Recall that for a rectifiable curve γ: [0,1] → ℂ, we let ‖γ‖ denote its length, i.e. ‖γ‖ = <Inline>1</Inline> |γ′(t)|dt, so that

$$ \left| {\int_{\gamma } {f(z)dz} } \right| \leqslant \left\| \gamma \right\| \cdot \left\| f \right\| $$

where ‖f‖ = sup{| f(z) |: z ∈ γ}. The reader is reminded that γ^ denotes the “physical curve”, that is the image of γ.