rectifiable curve

Suppose that is a collection of rectifiable curves in .

Suppose that there exists a rectifiable curve given by a function f(x).

Now let denote the set of rectifiable curves such that is non-rectifiable.

That is, these curves are a subset of the rectifiable curves.

The Cantor function is non-decreasing, and so in particular its graph defines a rectifiable curve.

These are not rectifiable curves, meaning they cannot be measured by being broken down into many segments approximating their respective lengths.

In 1902, Hurwitz published a short proof using the Fourier series that applies to arbitrary rectifiable curves (not assumed to be smooth).

Among the problems Dr. Alexander had solved was finding the polynomial hull of rectifiable curves in the space of several complex variables.

A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

In Geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.

In its simplest and original form, it asks under what conditions may a set E in two-dimensional Euclidean space be contained inside a rectifiable curve of finite length.

It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set.

It has a topological dimension of 1, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch Snowflake is infinite.

By using the Cauchy integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around a.

F. and M. Riesz Theorem: If is a simply connected planar domain bounded by a rectifiable curve (i.e. if ), then harmonic measure is mutually absolutely continuous with respect to arc length: for all , if and only if .

There is an analogous problem in geometric measure theory which asks the following: under what conditions may a subset E of Euclidean space be contained in a rectifiable curve (that is, when is there a curve with finite length that visits every point in E)?

A posteriori, for E to be contained in a rectifiable curve Γ, since Γ has tangents at H-almost every point in Γ (where H denotes one-dimensional Hausdorff measure), E must look flat when you zoom in on points in E.

The circle γ can be replaced by any closed rectifiable curve in U which has winding number one about a. Moreover, as for the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on its closure.