Recall that z(t) = x(t) + iy(t) for t \in [a,b]. The parameter t can be thought of as time.
Definition. A Jordan arc (or simple arc) does not intersect itself. That is, z(t_1) \ne z(t_2) for t_1 \ne t_2.
Definition. A Jordan curve (or simple closed curve) is a Jordan arc that has the property z(a) = z(b).
Example 1: z = \begin{cases}t + it & 0 \le t \le 1 \\ t + i & 1 < t \le 2\end{cases} is a simple arc, whose trace is the graph of the points. The arc would be traced out with a ‘speed’ of \sqrt2 between 0 and 1 because it covers a distance of \sqrt2 in 1 time unit.
Example 2: z = z_0 + Re^{i\theta} for 0 \le \theta \le 2\pi is an arc whose trace is a circle, centred at z_0 of radius R.
Example 3: z = z_0 + Re^{-i\theta} for 0 \le \theta \le 2\pi traces the same circle, but in the opposite direction. We use a negative in the exponent to allow the parameter to be increasing (fitting the time analogy).
Example 4: z = z_0 + Re^{2i\theta} for 0 \le \theta \le 2\pi again has the same trace, but it “covers” the circle twice.
In these examples, 2 and 3 are Jordan curves and 4 is not.
Definition. An arc/curve is called differentiable if z'(t) exists (at all t \in (a,b) for an arc, and at t \in [a,b] for a curve).
Definition. If z' exists and is continuous, then \int_a^b |z'(t)|\,dt exists and defines the arc length.
This is crucial because the length of an arc does not depend on the particular parametrisation. More specifically, if z(t) is any parametrisation of the image arc, we can define another one by t = \Phi(\tau) with \Phi(\alpha) = a and \Phi(\beta) = b such that \Phi \in C([\alpha, \beta]) and \Phi' \in C((\alpha, \beta)). Then, z(t) = Z(\tau) = z(\Phi(t)).
We will prove that the arc length is the same. Assume \Phi(\tau) > 0 for all \tau (that is, we always move forwards in time). Then, \begin{aligned} \int_a^b|z'(t)|\,dt &= \int_\alpha^\beta |z'(\Phi(\tau))| \Phi'(\tau)\,d\tau \\ &= \int_\alpha^\beta \left|Z'(\tau)\right|\,d\tau \end{aligned} which implies arc length is independent of parametrisation.
Definition. A contour is an arc/curve/Jordan curve such that z is continuous and z is piecewise differentiable. Additionally, if initial and final values coincide and there are no other self-intersections, it is a simple closed contour.
Theorem (Jordan curve theorem). Any simple closed contour divides \mathbb C into three parts:
Although it seems obvious, this is actually more complex. Consider a Möbius strip. This would take about 8 lectures to prove, so we’ll trust Jordan on this one.
Remark: The theorem still holds if we remove the requirement that z is piecewise differentiable. This leads to very freaky things such as space-filling curves.
Given a contour C, a contour integral is written \begin{aligned} \int_C f(z)\,dz \quad \text{ or }\quad \int_{z_1}^{z_2} f(z)\,dz. \end{aligned} We can write the second expression if we know:
Suppose the contour C is specified by z(t) with z_1 = z(a) and z_2 = z(b), with a \le t \le b, and suppose f is piecewise continuous on C. Then (reminiscent of line integrals), \int_C f(z)\,dz = \int_a^b f(z(t))z'(t)\,dt.