What makes a locus in the complex plane a contour?

The question is pretty straightforward: What makes a locus in the complex plane a contour?

I read through the Wolfram MathWorld pages for both a contour and the Cauchy integral formula (the page about residue theorem was too lofty for me) but neither of these define any criteria.

I completely understand that the contour is the curve in the complex plane that is the domain of integration for a line integral of a complex-valued function. I am also under the impression that a locus must be closed to be a contour, but I am not sure.

• I believe I may have used tags complex-analysis and complex-integration inappropriately; I welcome corrections to this. – Chase Ryan Taylor Aug 17 '17 at 4:03
• In complex analysis you need : $\int_a^b f(z)dz$ and $\int_\gamma f(z)dz$ where $a,b \in \mathbb{C}$, $f$ is an analytic function, and $\gamma$ is a (piecewise) $C^1$ finite-length curve, sometimes infinite-length. A closed curve means $\gamma(0) = \gamma(1)$. Contour often means closed curve. In some sense, the concatenation of closed curves is still a closed curve. – reuns Aug 17 '17 at 4:04