Alphabeta MathHow 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.
Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries
A complex path is read as the planar path through as the Euclidean plane and as a normed real algebra: what the identification preserves. Its polygonal length and rectifiability are therefore those of Paths in , inscribed polygonal sums, arc length as their supremum, and rectifiability. By A path in is rectifiable exactly when every coordinate has bounded variation, it is rectifiable exactly when both and have bounded variation. Every piecewise- complex path is rectifiable, and A continuous piecewise- path is rectifiable and its length is the sum of the speed integrals over its pieces gives its length as the sum of the speed integrals over its smooth pieces, allowing corners and zero-speed pieces.
Depends on
- $\mathbb C=\mathbb R[x]/(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves
- Paths in $\mathbb{R}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability
- A path in $\mathbb{R}^n$ is rectifiable exactly when every coordinate has bounded variation
- A continuous piecewise-$C^1$ path is rectifiable and its length is the sum of the speed integrals over its pieces
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 105 results over 20 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)