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The winding number vanishes on the unbounded component of the complement of the trace

Statement

Let $γ$ be a closed complex contour with trace $γ^{*}$ and length $L (γ)$ . Then $C ∖ γ^{*}$ has exactly one unbounded connected component $U_{\infty}$ , and

$n (γ, p) = 0 for every p \in U_{\infty} .$

More precisely, if $R > 0$ satisfies $γ^{*} \subseteq {z : ∣ z ∣ \leq R}$ and $∣ p ∣ > R + L (γ) / (2 π)$ , then $p \in U_{\infty}$ and $n (γ, p) = 0$ .

Facts & Assumptions

Given: A closed complex contour $γ : [a, b] \to C$ .

[L1]

For a compact $K \subseteq C$ , the complement $C ∖ K$ has exactly one unbounded connected component $U_{\infty}$ , every other component is bounded, and ${z : ∣ z ∣ > R} \subseteq U_{\infty}$ whenever $R > 0$ satisfies $K \subseteq {z : ∣ z ∣ \leq R}$ (The complement of a compact plane set has exactly one unbounded connected component).

[L2]

The index $n (γ, \cdot)$ is constant on every connected component of $C ∖ γ^{*}$ (The winding number is constant on each connected component of the complement of the trace).

[L3]

The winding number of a closed complex contour about a point off its trace is an integer (The winding number of a closed contour is an integer).

[L4]

If $∣ f (z) ∣ \leq M$ on the trace of a rectifiable contour $γ$ , with $M \geq 0$ , then $∣ \int_{γ} f (z) d z ∣ \leq M L (γ)$ (ML estimate: a contour integral is bounded by a supremum bound times path length).

[L5]

$n (γ, p) = (2 π i)^{- 1} \int_{γ} d z / (z - p)$ (The winding number of a closed contour about a point off its trace).

[L6]

A compact subset of a metric space is closed and bounded (A compact subset of a metric space is closed and bounded); the continuous image of a compact subset is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); a closed bounded interval is compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[L7]

The connected component $C (x)$ is the union of all connected subsets containing $x$ (Connected components, quasicomponents, and totally disconnected spaces) and contains every connected subset containing $x$ (The components of a space are its maximal connected subsets, they partition it, and each of them is closed).

[L8]

A subset of a metric space is bounded when it is empty or contained in some ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

[L9]

$∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ for complex $z, w$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L10]

The integers form an ordered commutative ring and their canonical image in $R$ is discrete; in particular the only integer of modulus below $1$ is $0$ (The integers form a commutative ring, The integers form a totally ordered ring, Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

Proof

technique · direct

1.1givenL1L6L8

The trace $γ^{*}$ is the continuous image of a compact interval, hence compact by [L6], and bounded by [L6], so there is $R > 0$ with $γ^{*} \subseteq {z : ∣ z ∣ \leq R}$ . By [L1] the set $C ∖ γ^{*}$ has exactly one unbounded component $U_{\infty}$ , and ${z : ∣ z ∣ > R} \subseteq U_{\infty}$ .

2.1step 1.1L4L5L9

Let $∣ p ∣ > R$ . For $w \in γ^{*}$ one has $∣ w ∣ \leq R$ , so $∣ w - p ∣ \geq ∣ p ∣ - ∣ w ∣ \geq ∣ p ∣ - R > 0$ by [L9]; hence $p \notin γ^{*}$ and $∣ 1 / (z - p) ∣ \leq 1 / (∣ p ∣ - R)$ on the trace. By [L4] and [L5], $∣ n (γ, p) ∣ \leq L (γ) / (2 π (∣ p ∣ - R))$ .

3.1step 1.1step 2.1L3L10

If in addition $∣ p ∣ > R + L (γ) / (2 π)$ then $∣ p ∣ - R > L (γ) / (2 π)$ , so step 2.1 gives $∣ n (γ, p) ∣ < 1$ ; since $n (γ, p)$ is an integer by [L3], it is $0$ by [L10]. Such $p$ exist, for instance $p = R + L (γ) / (2 π) + 1$ , and each lies in $U_{\infty}$ by step 1.1.

4.1step 3.1L2L7∎

By [L2] the index is constant on the connected component $U_{\infty}$ , and step 3.1 exhibits a point of $U_{\infty}$ where its value is $0$ ; hence $n (γ, p) = 0$ for every $p \in U_{\infty}$ , which by [L7] contains every connected unbounded subset of $C ∖ γ^{*}$ that meets it.

Depends on

Dependency tree · 2 levels

Sources

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.1, Properties (i) and (ii)