Curve and contour

A curve is described by a parametrization \(\gamma(t)=x(t)+iy(t)\): each value of the parameter \(t\) gives a point of the plane and, as \(t\) grows, the point moves and leaves a directed trajectory. The same trajectory has many parametrizations; for example \(\gamma(t)=(1-t)+it\) with \(0\le t\le1\) and \(\tilde\gamma(s)=(1-s^2)+is^2\) with \(0\le s\le1\) trace the same segment from \(1\) to \(i\).

The derivative \(\gamma'(t)=x'(t)+iy'(t)\) is the velocity vector. Requiring it to be continuous and nonzero guarantees a tangent line at every point and rules out cusps and stops. The length of the curve is \(L=\int_a^b|\gamma'(t)|\,dt\); for \(\gamma(t)=e^{it}\), \(0\le t\le2\pi\), it gives \(2\pi\).

A contour allows corners: it is a sequence \(\gamma_1,\dots,\gamma_n\) of smooth curves where the end of each is the start of the next. The boundary of a rectangle, with four sides, is the typical example. It is called closed if the last point equals the first, and simple if it does not cross itself, except at the endpoints of a closed contour.

Every curve has an opposite \(-\gamma\), which traces the same points in the reverse direction, for example \(-\gamma(t)=\gamma(a+b-t)\). Simple closed contours are positively oriented (counterclockwise) by convention. In the next concept, the contour integral will use exactly \(\gamma\) and \(\gamma'\) to turn an integral over the path into an ordinary real integral.