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Rectifiable curve

A rectifiable curve is a curve which has a well-defined finite length. Rectifiable curves are mainly important in complex analysis because they are needed to define the path integral.

Suppose γ : [a, b] -> C is a continuous function from an interval into the complex plane. This curve γ is called rectifiable if the following supremum is finite:

sup {∑i=1n |γ(ti)-γ(ti-1)| : n in N and at0<t1<...<tnb}
The value of this supremum is called the length of the curve γ.

In an analogous manner (by replacing the absolute value with the Euclidean distance or a norm), one can define rectifiable curves γ : [a, b] -> Rn and, more generally, γ : [a, b] -> V where V is a normed vector space.

Every continuous and piecewise continuously differentiable[?] curve γ is rectifiable, and its length can be computed as the ordinary Riemann integral

L = ∫ab |γ'(t)| dt



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