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A primitive of a complex function on an open set
Definition
Let be open and let . A primitive of on is a holomorphic function in the sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions such that for every .
Depends on
Used by
- Cauchy's theorem on a star-shaped domain: every closed rectifiable contour integral of a holomorphic function is zero Corollary
- The integral of a continuous derivative over a cycle is zero Corollary
- FALSE: every continuous complex-valued function on a convex domain has a primitive False statement
- On a convex open set the difference quotient is an average of the derivative along the segment Lemma
- Star-shaped plane domains are homologically simply connected Proposition
- Vanishing integrals around triangles construct a primitive for a continuous function on a star-shaped domain Proposition
- A continuous function holomorphic away from one point on a star-shaped domain has a primitive and zero closed-contour integrals Theorem
- A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
- Equivalent characterisations of a homologically simply connected domain Theorem
- Every holomorphic function on a homologically simply connected domain has a primitive Theorem
- The integral of dz/(z-p) along a contour is the increment of a continuous logarithm Theorem
- The line integral of a continuous function admitting a primitive is that primitive's endpoint increment along every rectifiable path Theorem
Dependency tree · two levels
4 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
- E. Stein and R. Shakarchi, Complex Analysis, Ch. 1, §3 (standard reference, not scraped)