Alphabeta Math

Contour Integration — Examples

About this page
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • 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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 12 results · all verified · 5 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Complex Analysis
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

An exponential contour integral approximated by Riemann sums and evaluated by parametrization and a primitive

Example

Integrating a complex polynomial along a segment and a parabola by a primitive and by parametrization

Example

The integral of complex conjugation from -1 to 1 differs along a semicircle and a polygonal path

Example

ML bounds for rational integrands on a semicircular arc and a line segment

Example

Direct computation of the integral of 1/(z-a) around a semicircle and a full circle centred at a

Example

The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration

Example

Assembling a keyhole contour from two radial segments and two circular arcs

Example

The rectifiable Riemann–Stieltjes definition on an explicit polygonal contour with corners

Counterexample

Reversing orientation does not preserve a complex contour integral

False statement

FALSE: the modulus of a contour integral always equals the absolute line integral

False statement

FALSE: contour length depends only on the trace and ignores multiplicity

False statement

FALSE: parametrization independence makes orientation reversal leave every contour integral unchanged

Sources