ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The integral of 1 / (1 + x^2) over the real line is pi

Example

$\int_{- \infty}^{\infty} \frac{d x}{1 + x^{2}} = π .$

Facts & Assumptions

Given: The rational function $R (z) = 1 / (1 + z^{2})$ .

[L1]

A rational function with no real poles and a two-degree denominator gap is evaluated by the residues of its upper-half-plane poles (Rational improper integrals without real poles are upper-half-plane residue sums).

Verification

technique · computation

1.1

The only pole of $R$ in the upper half-plane is $i$ , and it is simple with $Res (R, i) = \lim_{z \to i} \frac{z - i}{(z - i) (z + i)} = \frac{1}{2 i} .$

givenalgebra

2.1

Applying [L1] gives $\int_{- \infty}^{\infty} \frac{d x}{1 + x^{2}} = 2 π i \cdot \frac{1}{2 i} = π .$

step 1.1L1∎

Depends on

Rational improper integrals without real poles are upper-half-plane residue sums

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Howell and J. Mathews, Complex Analysis, Ch. 8 §8.3 (standard reference, not scraped)