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:

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 metric space.

Every continuous and piecewise continuously differentiable curve γ : [a, b] -> C is rectifiable, and its length can be computed as the ordinary Riemann integral