Index of a Path. The Argument Principle (Continued). Rouché’s Theorem. Theorem 1.1 Revisited. Proof of Theorem 3.2. The Maximum Modulus Principle. Proof of Theorem 3.3

Abstract

We continue to progress towards the standard version of the Argument Principle.

By Example 6.1, Theorem 9.1, and Proposition 16.2, for a closed path $\gamma\ \mathrm{in}\ \mathbb{C}\backslash\{0\}$ the expression

$$ \frac{1}{2\pi i} \int\nolimits_{\gamma} \frac{1}{z} dz $$

(17.1)

is an integer, namely, the integer m supplied by Proposition 16.2.