From points to roads
Through this rung we have watched a complex function act on the plane: the image of a region under a translation, a rotation, the squaring map, inversion. But all of that picturing secretly relied on one humbler object you have only glimpsed in passing — the curve. When you sketched what z^2 does to the unit circle, or how 1/z bends a grid, you were really tracking what the map does to a road drawn through the z-plane. Curves are the true canvas of this subject, and this final guide in the rung puts them centre stage.
A curve in the plane is described by letting a real parameter t run over an interval [a, b] and writing z(t) = x(t) + i y(t). As t sweeps from a to b, the point z(t) traces a road through the plane — this is a parametrized path. The arrow of increasing t even gives the road a direction, a sense of travel from a starting point z(a) to an ending point z(b). One real input, one moving complex output: a curve is the simplest possible complex-valued function of a real variable, and that simplicity is exactly why it is the right object to push a map through.
What a map does to a road
Here is the move that turns the mapping viewpoint into something you can actually draw. Take a path z(t) in the input plane and feed every one of its points through a continuous function f. Out comes a brand-new path w(t) = f(z(t)) in the output plane. The map carries the whole road across, point by point, and because f is continuous the image road never tears — nearby points on the input curve stay nearby on the output curve. This is exactly the payoff of continuity you met two guides ago, now doing visible work.
Work it through once and it sticks. Send the unit circle z(t) = e^(i t), for t in [0, 2 pi), through the squaring map. Then w(t) = (e^(i t))^2 = e^(i 2 t): as t runs once around, the argument 2t runs twice around, so the image is the unit circle traversed TWICE. The doubling-of-angle behaviour you saw abstractly in guide two becomes a concrete, countable fact about a road — the map winds the circle around the origin two full times.
z(t) = e^(i t), t in [0, 2 pi) input: unit circle, once around f(z) = z^2 w(t) = f(z(t)) = (e^(i t))^2 = e^(i 2 t) as t: 0 -> 2 pi the angle 2t: 0 -> 4 pi => image is the unit circle traversed TWICE
Two real coordinate-roads, woven together
The f = u + i v split that has steadied us all rung long steadies us here too. A path w(t) = f(z(t)) is itself complex, so it breaks into a real coordinate u(t) and an imaginary coordinate v(t), each an ordinary real function of the real parameter t. Drawing the image curve is then nothing more exotic than plotting the planar point (u(t), v(t)) as t advances — a parametric curve of precisely the kind your earlier calculus already taught you to trace.
Let one concrete example tie the whole rung together. Take the vertical line z(t) = 1 + i t in the input plane and square it: w = (1 + i t)^2 = (1 - t^2) + i (2 t), so u = 1 - t^2 and v = 2 t. Eliminate t with t = v/2, and u = 1 - v^2/4 — a sideways parabola. A straight road in the z-plane becomes a curved road in the w-plane, and you can name the curve exactly by working with u and v alone. That is the mapping viewpoint, limits, continuity, and the u-plus-iv bridge all cashing out in a single picture.
Make this your reflex. When a map mystifies you, stop staring at the formula and feed it a road. Parametrize one simple test curve — a grid line, a ray from the origin, a circle — push it through, separate into u(t) and v(t), and eliminate t. The shape of the image curve usually tells you more about the map in one stroke than pages of algebra, and it works no matter how forbidding the formula looks.
Which roads, and which regions, we will trust
Not every road is equally well-behaved, and the theory ahead is fussy about which it admits. A simple curve never crosses itself; a closed curve returns to where it began, so z(a) = z(b); a simple closed curve does both, drawing a single loop with a clean inside and outside. And to do calculus on a curve we usually ask it to be smooth — built from pieces whose velocity z'(t) exists and never stalls to zero. These are not pedantic labels; each condition is there because some theorem to come will lean on it.
Curves also let us speak precisely about regions, which is why this guide quietly upgrades the loose word "region" from guide one. A domain is simply connected when every closed loop drawn inside it can be shrunk continuously to a point without ever leaving the domain — an open disk qualifies, but a disk with a puncture at its centre does not, because a loop circling the hole snags on it. Hold onto this distinction with both hands: it is the single hypothesis that will decide whether an integral around a closed loop comes out to zero or not.
The roads ahead
Here is where the title earns itself. Everything you have built — mappings, limits, continuity, the u-plus-iv split, and now curves — converges on the two great themes that the rest of complex analysis is made of. The first is the derivative. The next rung asks for the limit of the difference quotient (f(z) - f(z_0))/(z - z_0) as z approaches z_0 along ANY road, and demands that all roads agree. That single demand is complex differentiability, and it is incomparably stronger than its real cousin: one complex derivative forces a function to be infinitely differentiable and to equal its own power series — a rigidity with no real-variable analogue whatsoever. Do not let anyone sell it to you as "the same limit as in calculus."
The second great theme is the one curves were quietly preparing us for all along: integration. Instead of integrating over an interval, complex analysis integrates ALONG a path — a contour integral, written informally as the sum of f(z) dz collected as z travels down the road. This is why the direction and traversal of a path, flagged earlier as 'extra data,' turn out to be load-bearing: reverse the road and the integral flips sign. From this single idea blooms the Cauchy theory — that for a well-behaved function on a simply connected domain, the integral around any closed loop vanishes — and from there residues, the evaluation of stubborn real integrals, and most of the subject's power.