ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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The unit circle traversed three times has index $3$ at every interior point

Example

Let $γ (t) = \exp (3 i t)$ for $t \in [0, 2 π]$ . Then $γ$ is a closed complex contour with trace the unit circle ${∣ z ∣ = 1}$ , and

$n (γ, z) = 3 for ∣ z ∣ < 1, n (γ, z) = 0 for ∣ z ∣ > 1.$

The function $λ (t) = 3 i t$ is a continuous logarithm of $γ - 0$ along $γ$ , its imaginary part $θ (t) = 3 t$ is a continuous argument running from $0$ to $6 π$ , and $(θ (2 π) - θ (0)) / (2 π) = 3$ recovers the index at the origin.

Facts & Assumptions

Given: The contour $γ (t) = \exp (3 i t)$ on $[0, 2 π]$ .

[L1]

For $a \in C$ , $r > 0$ and $k \in Z$ , the contour $γ_{k} (t) = a + r \exp (i k t)$ on $[0, 2 π]$ is a closed complex contour with $n (γ_{k}, z) = k$ for $∣ z - a ∣ < r$ and $n (γ_{k}, z) = 0$ for $∣ z - a ∣ > r$ ; for $k \neq 0$ its trace is ${z : ∣ z - a ∣ = r}$ (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

[L2]

For a closed complex contour $γ$ , a point $p$ off its trace and a continuous argument $θ$ of $γ - p$ along $γ$ , $n (γ, p) = (θ (b) - θ (a)) / (2 π)$ (The winding number is the increment of a continuous argument divided by $2 π$ ).

[L3]

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

[L4]

A continuous logarithm of $γ - p$ along $γ$ is a continuous $λ$ with $\exp (λ (t)) = γ (t) - p$ for every $t$ , and its continuous argument is $Im λ$ (Continuous logarithms and continuous arguments along a contour).

[L5]

For real $x, y$ , $∣ \exp (x + i y) ∣ = e^{x}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ); for $z = a + b i$ with $a, b$ real, $Im z = b$ (Real and imaginary parts, complex conjugation, and modulus).

Verification

technique · direct

1.1

Apply [L1] with $a = 0$ , $r = 1$ and $k = 3$ : the contour is $γ$ , it is a closed complex contour with trace ${∣ z ∣ = 1}$ , and $n (γ, z) = 3$ for $∣ z ∣ < 1$ while $n (γ, z) = 0$ for $∣ z ∣ > 1$ .

givenL1L3

1.2

The map $λ (t) = 3 i t$ is continuous on $[0, 2 π]$ and satisfies $\exp (λ (t)) = \exp (3 i t) = γ (t) - 0$ , so it is a continuous logarithm of $γ - 0$ along $γ$ in the sense of [L4], with continuous argument $θ (t) = Im (3 i t) = 3 t$ by [L5].

givenL4L5

2.1

The argument increment is $θ (2 π) - θ (0) = 6 π - 0 = 6 π$ , so [L2] gives $n (γ, 0) = 6 π / (2 π) = 3$ , the same value step 1.1 assigns at the interior point $0$ .

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.1 (standard reference, not scraped)

The unit circle traversed three times has index 33 at every interior point