| dbp:mathStatement
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- If is a rectifiable Jordan curve in and if is a continuous mapping holomorphic throughout the inner region of , then the integral being a complex contour integral. (en)
- Assume is a rectifiable, positively oriented Jordan curve in the plane and let be its inner region. For every positive real , let denote the collection of squares in the plane bounded by the lines , where runs through the set of integers. Then, for this , there exists a decomposition of into a finite number of non-overlapping subregions in such a manner that
Each one of the subregions contained in , say , is a square from .
Each one of the remaining subregions, say , has as boundary a rectifiable Jordan curve formed by a finite number of arcs of and parts of the sides of some square from .
Each one of the border regions can be enclosed in a square of edge-length .
If is the positively oriented boundary curve of , then
The number of border regions is no greater than , where is the length of . (en)
- Let be a rectifiable curve in and let be a continuous function. Then
and
where is the oscillation of on the range of . (en)
- Let be a rectifiable curve in the plane and let be the set of points in the plane whose distance from is at most . The outer Jordan content of this set satisfies . (en)
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