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 cos x / (1 + x^2) over the real line is pi / e

Example

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

Facts & Assumptions

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

[L1]

For $λ > 0$ , the Fourier integral of $e^{i λ x} R (x)$ is the residue sum of $e^{i λ z} R (z)$ in the upper half-plane (Rational Fourier integrals are evaluated by residues and Jordan's lemma).

Verification

technique · computation

1.1

Apply [L1] with $λ = 1$ . The only upper-half-plane pole is $i$ , and $Res ⁣ (\frac{e^{i z}}{1 + z^{2}}, i) = \frac{e^{i i}}{2 i} = \frac{e^{- 1}}{2 i} .$

givenL1algebra

2.1

Hence $\int_{- \infty}^{\infty} \frac{e^{i x}}{1 + x^{2}} d x = 2 π i \cdot \frac{e^{- 1}}{2 i} = \frac{π}{e} .$ The value is real, so taking real parts yields $\int_{- \infty}^{\infty} \frac{\cos x}{1 + x^{2}} d x = \frac{π}{e} .$

step 1.1algebra∎

Depends on

Rational Fourier integrals are evaluated by residues and Jordan's lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 3 §2.1 (standard reference, not scraped)