CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The normalized integral around a positively oriented circle centred at a is 1

Statement

For a positively oriented circle $γ (t) = a + r \exp (i t)$ with $r > 0$ , $\frac{1}{2 π i} \int_{γ} \frac{d z}{z - a} = 1.$

Facts & Assumptions

Given: A positively oriented circle of positive radius centred at $a$ .

[L1]

The integer-monomial circle formula gives $\int_{γ} (z - a)^{- 1} d z = 2 π i$ (On a positively oriented circle about a, the integral of (z-a)^m is zero for every integer m except -1, and is 2 pi i for m=-1).

[L2]

The complex numbers form a field containing the embedded real field, every complex number has a unique form $a + b i$ , and every nonzero element has an inverse ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ).

[L3]

The real number $π$ is positive (Pi as twice the smallest positive zero of cosine).

Proof

technique · direct

1.1

By [L3], $2 π \neq 0$ in the embedded real field; the unique complex-coordinate form in [L2] then gives $2 π i \neq 0$ , so division is licensed.

L2L3algebra

2.1

Substitute [L1] and divide to obtain $1$ . The value is independent of the positive radius, and no winding-number or Cauchy theorem is used.

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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