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(t_i)-γ(t_i-1)| : n in N and a≤t₀<t₁<...<t_n≤b}

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 = ∫_a^b |γ'(t)| dt