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.
Simple poles
Definition
An isolated singularity is a simple pole when it is a pole of order in the sense of Isolated singularities: removable, poles, and essential singularities.
Remarks
Equivalently, is a simple pole of when extends holomorphically across and takes a nonzero value there.
Depends on
Used by
- The one-variable function 1 / z does not extend across the origin Counterexample
- An indented arc around a simple singularity contributes the expected residue fraction Lemma
- At a simple pole the residue is the limit of (z-a)f(z) Lemma
- Characterizations of poles Theorem
- Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions Theorem
- Divisor and residue laws for elliptic functions Theorem
Dependency tree · one level
1 result within one dependency step 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
- Jean-Baptiste Campesato, MAT334 course page and notes index (standard reference, not scraped)