From graph to motion
In the previous guide we adopted the mapping viewpoint: a complex function is not a graph but a machine that picks up the whole z-plane and sets it down in a w-plane. Now we put that machine to work on the small handful of functions that everything else is built from. The plan is simple — take a familiar shape (a line, a circle, a grid), feed it in, and watch what comes out the other side.
One bookkeeping habit makes every picture computable. Write the input as z = x + i y and the output as f = u + i v, where u(x, y) and v(x, y) are two real functions of two real variables. A single complex rule therefore secretly carries two real rules, and to find them you just compute f(x + i y) and separate the real and imaginary pieces. This is the bridge back to the multivariable calculus you already know.
f(z) = u(x, y) + i v(x, y) f(z) = z^2 : u = x^2 - y^2 , v = 2 x y f(z) = z-bar : u = x , v = -y f(z) = 1/z : u = x/(x^2+y^2), v = -y/(x^2+y^2)
The gentle maps: slide, turn, mirror
Start with the calmest motion. The translation z + a picks up every point and slides it by the same vector a — nothing rotates, nothing stretches, nothing flips. A circle of radius 3 about the origin becomes an identical circle about a; a square slides over unchanged. Translations are rigid motions, and they have no fixed point unless a = 0.
Multiplying by a constant, the map a z, does two things at once and is the geometric meaning of complex multiplication itself. Write a = r e^(i alpha); then a z multiplies every modulus by r (a stretch or shrink) and adds alpha to every argument (a rotation about the origin). Multiplying by i, for instance, is a clean 90-degree turn, since i = e^(i pi/2). Because the stretch factor and turn angle are the same everywhere, a z is the prototype of a conformal map — it preserves the angles between curves.
Now a sly one. The reflection z-bar flips the plane across the real axis like a mirror: x + i y goes to x - i y. It preserves distances and the sizes of angles, so it looks almost as tame as a rotation — but it reverses the sense of angles, turning a counter-clockwise sweep into a clockwise one. Keep z-bar in mind: it is the cleanest example of a map that is perfectly smooth as a real map of the plane yet not complex-differentiable anywhere, a distinction the next two guides will sharpen.
The benders: z^2 and 1/z
The squaring map z^2 is the first function that genuinely bends the plane, and polar form makes it transparent: if z = r e^(i theta) then z^2 = r^2 e^(i 2 theta). It doubles every angle and squares every distance. The striking consequence is that it wraps the plane twice around itself — let z sweep the upper half-plane (angles 0 to pi) and the image already covers the whole w-plane (angles 0 to 2 pi). Run over the entire plane and every non-zero target is hit twice, since z and -z share a square.
The inversion 1/z turns the plane inside out: in polar form 1/z = (1/r) e^(-i theta), so it inverts distance (far points come close, near points fly out, the unit circle stays put) and reflects across the real axis. Its signature is that it sends lines and circles to lines and circles, once you agree to treat a line as a circle through infinity. A vertical line can become a circle through the origin; the inside and outside of the unit circle swap places.
There is one honest snag: 1/z is undefined at z = 0, so as a plane-to-plane map its domain excludes the origin. The clean repair is to adjoin a point at infinity and work on the Riemann sphere, where 1/z swaps 0 and infinity and becomes a tidy one-to-one map of the whole sphere. Together, slide, rotate-scale, and invert are the building blocks of every Mobius transformation you will meet later.
Reading a map by hand
These pictures are not magic; you compute them. To see what a map does to a curve, push the curve through the rule and simplify. Here is the routine that turns 'I wonder where this goes' into a definite answer, worked on the curve Re(z) = 1/2 under 1/z.
- Name the image variable. Set w = f(z), so here w = 1/z, and invert the rule to express z in terms of w: z = 1/w.
- Write the curve as an equation in z. The line is Re(z) = 1/2, that is x = 1/2.
- Substitute z = 1/w and split into real and imaginary parts. With w = s + i t, you get z = (s - i t)/(s^2 + t^2), so Re(z) = s/(s^2 + t^2).
- Impose the equation and simplify. Setting s/(s^2 + t^2) = 1/2 gives s^2 + t^2 = 2s, i.e. (s - 1)^2 + t^2 = 1 — the circle |w - 1| = 1, which passes through the origin.
- Sanity-check a point. The point z = 1/2 (on the line) maps to w = 2, which sits on that circle. The picture and the algebra agree.
The same recipe handles z^2 if you work in polar form instead: a circle |z| = R maps to |w| = R^2, and a ray arg z = theta maps to the ray arg w = 2 theta. Choosing the right coordinates — Cartesian for 1/z, polar for z^2 — is half the skill.
Approaching from every direction
Watching maps move shapes also sharpens what a complex limit must mean. Saying f(z) tends to L as z tends to z_0 is the familiar epsilon-delta idea — every target disk of radius epsilon around L can be reached by taking z within some disk of radius delta around z_0 — measured by the modulus |z - z_0| and |f(z) - L|.
But |z - z_0| < delta carves out a whole disk, not an interval. So z may approach z_0 from any direction — along the real axis, along the imaginary axis, along a spiral — and the same single L must be reached on every route. On the real line you had only left and right to reconcile; in the plane there are infinitely many roads in, and the answer must not depend on which one you take. If two approaches disagree, the limit simply does not exist.
Once limits behave, continuity follows in the obvious way, and it splits along the u + i v line: f is continuous at a point exactly when both u and v are continuous there as ordinary real functions. That clean equivalence — and the curves and paths these maps act on — are the threads the rest of this rung picks up. Here the lesson is geometric: you now know, by sight, what the five basic maps do to the plane.