Mapping the Complex Plane

Complex Analysis · Axiom Academy

A complex function isn't a curve you graph — it's a transformation you watch move one plane into another. You can't graph a complex function — so you watch it move A real function f(x) has a one-number input and a one-number output, so its graph fits on a flat page. A complex function w = f(z) eats a point z = x + iy and spits out another point w = u + iv — two numbers in, two numbers out. A full graph would need four dimensions. So we give up on graphing it and do something better: we lay an input plane and an output plane side by side and watch how f maps one onto the other. Press play. A point and a little patch of grid sit in the left z-plane; the simplest map, f(z) = z + 1 , sends each one to its image in the right w-plane. Adding 1 nudges every point one unit right — the whole patch slides over, unbent. Each image lands exactly where the arithmetic w = z + 1 puts it. A translation is the gentlest map of all: every point of the plane just slides — the patch keeps its shape, only its address changes. Drag the input — watch the output obey w = z^ 2 Grab the blue dot z in the left plane and move it anywhere. Every instant, the right plane shows w = z^ 2 — computed from wherever you put z , using . Drive z around a circle and you'll see w race around twice as fast: squaring doubles the angle and squares the distance from the origin. The output isn't drawn in by hand — it's the literal arithmetic of z^ 2 at the point you're holding. Bend an entire grid through w = z^ 2

This is the written version of the interactive lesson above. See the full Complex Analysis course.