Piecewise Smooth Curves in the Complex Plane

Piecewise Smooth Curves in the Complex Plane

We will soon look at integrals of complex functions along piecewise smooth curves in the complex plane but we will first need to properly define what we mean by "curve" and "piecewise smooth curves".

 Definition: A continuous function $\gamma : [a, b] \to \mathbb{C}$ is called a Curve in the complex plane. The point $\gamma (a)$ is called the Initial Point of the curve, and the point $\gamma (b)$ is called the Terminal Point of the curve.

For example, consider the following curve $\gamma : [0, 2\pi] \to \mathbb{C}$ defined by:

(1)
\begin{align} \quad \gamma (t) = e^{it} = (\cos t + i \sin t) \end{align}

Then $\gamma$ traverses the circle centered at the origin with radius $1$ whose initial and terminal point is $\gamma (0) = \gamma (2\pi) = 1 + 0i$, in a counterclockwise direction. For another example, consider the curve $\gamma : [0, 1] \to \mathbb{C}$ defined by:

(2)
\begin{align} \quad \gamma (t) = t + i \end{align}

Then $\gamma$ traverses the horizontal line segment with initial point $0 + i$ and terminal point $1 + i$.

Of course, we can also consider vertical line segments as curves. For example, consider the curve $\gamma : [0, 1] \to \mathbb{C}$ defined by:

(3)
\begin{align} \quad \gamma (t) = 1 + ti \end{align}

Then $\gamma$ traverses the vertical line segment with initial point $1 + 0i$ and terminal point $1 + i$.

We will be wanting to look at particular types of curves in the succeeding sections, so we will now make some important definitions which classify curves in the complex plane.

 Definition: A curve $\gamma : [a, b] \to \mathbb{C}$ in the complex plane is said to be a Piecewise Smooth Curve if there exists finitely many points $a = a_0, a_1, ..., a_n = b$ with $a = a_0 < a_1 < ... < a_n = b$ called a Partition of $[a, b]$, for which: 1) $\gamma$ is infinitely differentiable on each open subinterval $(a_{k}, a_{k+1})$. 2) The derivative of $\gamma$ on each closed subinterval $[a_k, a_{k+1}]$ are continuous.

(a) and (b) simply state that a piecewise smooth curve is a curve made up of finitely many smooth pieces. Pretty much all of the curves that we will encounter will be piecewise smooth curves and in general, it is not hard to distinguish between a curve that is piecewise smooth and one that is not if we have an idea of what the graphs of these curves look like.