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.
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Rectifiable complex contours, reversal, concatenation, closedness, and orientation
Definition
A complex contour is a rectifiable path in the sense of Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries. It is closed when . Its reversal is .
If satisfy , their concatenation is defined by the same two affine pieces as in Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations. An increasing continuous bijection of compact parameter intervals preserves orientation; a decreasing one reverses orientation. The underlying length is unchanged by either monotone reparametrization by Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal.
Depends on
- Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries
- Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations
- Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal
Used by
- The integral of a continuous complex derivative over every closed rectifiable contour is zero Corollary
- The absolute line integral over a rectifiable path using its arc-length function Definition
- The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral Definition
- Assembling a keyhole contour from two radial segments and two circular arcs Example
- Complex line integrals change sign under reversal and add under concatenation Proposition
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 32 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- E. Stein and R. Shakarchi, Complex Analysis, Ch. 1, §3 (standard reference, not scraped)