Complex Analysis/rectifiable curve
Definition
[edit | edit source]Let be a continuous curve. It is called rectifiable if its length
is finite, and is called the length of .
Approximation of path length by polygonal chain
[edit | edit source]The following image shows how a polygonal chain can be used to approximate the length of a curve .
Estimation of length
[edit | edit source]The length of the polygonal chain underestimates the actual length of a rectifiable curve , i.e. . In general, . By applying the triangle inequality, we get if the path's trace is not a line.
Path length for differentiable paths
[edit | edit source]If is continuously differentiable, then is rectifiable. Let , then there existsmean value theorem such that
Riemann sum as length of polygonal chain
[edit | edit source]The right-hand side of the above equation for the polygonal chain is a Riemann sum for the integral . If we take the maximum of the interval widths for to infinity, the length of the polygonal chains converges to the length of the path
Length for continuously differentiable paths
[edit | edit source]Let be a continuously differentiable path, then
gives the length of the path .
Note - Length for continuously differentiable paths
[edit | edit source]Since is continuously differentiable, is a continuous function. Since is a compact interval, takes a minimum and maximum. Therefore, and are bounded, and we have:
Piecewise continuously differentiable curves
[edit | edit source]In general, piecewise -curves are always rectifiable, because we can apply the above considerations to the individual parts of the curve, which then additively give the length of the entire curve. In the further course of complex analysis, paths (e.g. over the triangle edge) are considered that only possess the property of continuous differentiability in a piecewise manner, for which we can then still calculate the length as the sum of the arc lengths.
Non-rectifiable curve
[edit | edit source]As an example of a non-rectifiable curve, consider ,
Continuity - continuous differentiability
[edit | edit source]First, is continuous and, on each interval , even continuously differentiable. On these intervals, the length is given by
Calculation of improper integral
[edit | edit source]For , this converges to
so is not rectifiable.
See also
[edit | edit source]
Page Information
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Translation and Version Control
[edit | edit source]This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:
- Source: rektifizierbare Kurve - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve
- Date: 12/11/2024