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Complex Analysis/rectifiable curve

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Definition

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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

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The following image shows how a polygonal chain can be used to approximate the length of a curve .

rectifiable curve - approximation of length by polygonal chain - created with Geogebra on Linux

Estimation of length

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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

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If is continuously differentiable, then is rectifiable. Let , then there existsmean value theorem such that

Riemann sum as length of polygonal chain

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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

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Let be a continuously differentiable path, then

gives the length of the path .

Note - Length for continuously differentiable paths

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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

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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

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As an example of a non-rectifiable curve, consider ,

Continuity - continuous differentiability

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First, is continuous and, on each interval , even continuously differentiable. On these intervals, the length is given by

Calculation of improper integral

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For , this converges to

so is not rectifiable.

See also

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Page Information

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Translation and Version Control

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