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The integral of a continuous complex derivative over every closed rectifiable contour is zero
Statement
If is holomorphic on an open set, is continuous there, and is a closed rectifiable contour in that set, then .
Facts & Assumptions
Given: A function and a closed rectifiable contour as in the Statement.
The contour fundamental theorem gives (The line integral of a continuous function admitting a primitive is that primitive's endpoint increment along every rectifiable path).
A contour is closed exactly when its endpoint values agree (Rectifiable complex contours, reversal, concatenation, closedness, and orientation).
Proof
Apply [L1] and then [L2]: the endpoint increment is .
Thus the integral vanishes, including for a constant closed contour.
Depends on
Used by
- Cauchy's theorem on a star-shaped domain: every closed rectifiable contour integral of a holomorphic function is zero Corollary
- A nonvanishing holomorphic function on a domain with no holomorphic logarithm Counterexample
- FALSE: every continuous complex-valued function on a convex domain has a primitive False statement
- FALSE: every continuous complex-valued function on a domain has a primitive False statement
- A continuous function holomorphic away from one point on a star-shaped domain has a primitive and zero closed-contour integrals Theorem
- Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain Theorem
Dependency tree · two levels
16 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)