Alphabeta MathHow statement and proof provenance work
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Function elements and direct analytic continuation
Definition
A function element is a pair where is a complex domain (A complex domain is a nonempty connected open subset of ) and is holomorphic (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).
Let and be function elements. We say that is a direct analytic continuation of when there is a point such that the germs and agree (Holomorphic germs at a point).
Because equality of germs is symmetric, direct analytic continuation is a symmetric relation on function elements. It records local agreement on an overlap, not inclusion of one domain in the other.
Depends on
Used by
- Analytic continuation along a path by admissible chains Definition
- Singular boundary points and natural boundaries of function elements Definition
- The germ space of a complete analytic function Definition
- Schwarz reflection is an analytic continuation construction Remark
- The terminal germ of a continuation along a fixed path is chain-independent Theorem
Dependency tree · two levels
7 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
- Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 8 §§1.3-1.4 (standard reference, not scraped)
- Curtis T. McMullen, Riemann Surfaces, Ch. 4 (standard reference, not scraped)