curves, paths and arcs in the plane

A curve in the plane is the trail traced by a moving point. We describe it with a parametrization: a continuous function z(t) = x(t) + i y(t) that, as the real parameter t runs over an interval [a, b], plots out the curve as a path in the complex plane. The point z(a) is the start, z(b) is the end. Thinking of t as time, z(t) is the position of a particle drifting through the plane, and the curve is its trajectory.

A few standard distinctions tidy this up. An arc is a curve given by such a continuous z(t). It is a simple arc if it never crosses itself (z(t) hits no value twice). It is a closed curve if it returns to where it began, z(a) = z(b); a simple closed curve (a Jordan curve) is a closed loop with no self-crossings, like a circle or a polygon. A smooth curve has a continuous, non-zero derivative z'(t) — a well-defined tangent direction with no corners — and a path or contour is typically a chain of finitely many smooth arcs joined end to end, allowing corners. For example z(t) = e^(i t), t from 0 to 2 pi, is the unit circle traversed once counter-clockwise, a simple closed smooth curve.

Curves are the stage for the second half of complex analysis. They are the objects functions act on (you push a curve through f and watch its image bend), and, crucially, they are the contours along which you integrate. The whole theory of contour integration, Cauchy's theorem, the residue theorem, and the argument principle is built on integrating functions along these paths — so a clear grip on parametrized curves, their orientation (direction of travel), and when one can be deformed into another is the foundation everything later stands on.