Rectifiable complex contours, reversal, concatenation, closedness, and orientation

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

• A closed contour path-homotopic to a constant loop has zero integral against every holomorphic function Corollary
• Goursat's theorem for rectangles: a holomorphic function integrates to zero around every rectangle contained in its domain Corollary
• The integral of a continuous complex derivative over every closed rectifiable contour is zero Corollary
• The winding number is the increment of a continuous argument divided by 2π Corollary
• Banach algebra valued contour integral Definition
• Complex chains, their traces, and cycles Definition
• Continuous logarithms and continuous arguments along a contour Definition
• Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter Definition
• 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
• The winding number of a closed contour about a point off its trace Definition
• A continuous argument computed along a spiralling contour Example
• Assembling a keyhole contour from two radial segments and two circular arcs Example
• The winding numbers of a keyhole contour about the origin and about an excluded point Example
• The winding number depends only on the trace of the closed contour False statement
• A contour missing a point subdivides into arcs lying in discs that miss it Lemma
• Index of the boundary of a graph-bounded plane region Lemma
• Tagged sums approximate a contour integral within oscillation times length Lemma
• Complex line integrals change sign under reversal and add under concatenation Proposition
• Reversal negates and concatenation adds winding numbers Proposition
• Conventions for chains, cycles and the homological adjective on this page Remark
• A contour integral of a jointly continuous, parameter-holomorphic integrand is holomorphic Theorem
• Chain integration and the index are additive in the chain, and reverse with it Theorem
• Endpoint-fixed homotopic paths have equal holomorphic line integrals Theorem
• The integral of dz/(z-p) along a contour is the increment of a continuous logarithm Theorem
• The winding number of a closed contour is an integer Theorem

Dependency tree · two levels

8 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)