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Admissible cycles for the residue theorem
Definition
Let be open, let be meromorphic on in the sense of Meromorphic functions on a plane domain, and let be its pole set. By Poles of a meromorphic function form a closed discrete set and are at most countable, is closed and discrete in .
A complex cycle is admissible for the residue theorem in when
- ,
- is null-homologous in in the sense of Null-homologous cycles and homologous cycles in an open set.
For such a pair the candidate residue sum is
Only points with nonzero index can contribute, and the next lemma shows that there are only finitely many of them.
Depends on
Used by
- Zero and pole counts weighted by multiplicity and winding number Definition
- FALSE: the residue theorem applies to every cycle in the ambient domain False statement
- Only finitely many singularities contribute to the residue sum of an admissible cycle Lemma
- Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions Theorem
- Divisor and residue laws for elliptic functions Theorem
- Residue theorem on a compact Riemann surface Theorem
- The residue theorem for a null-homologous cycle Theorem
Dependency tree · two levels
22 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
- L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §5.1 (standard reference, not scraped)