Transforming the Plane

Complex Analysis · Axiom Academy

Every complex function is a way of bending the plane — and the good ones bend it without ever breaking a right angle. A function that moves every point Stop reading a complex function w = f(z) as a graph. Read it as a machine that moves the plane : it grabs each point z and sets it down somewhere new, at w = f(z) . Feed it a whole picture — a grid, a region, a shape — and it hands you a warped copy. Watch a plain square grid get carried over by w = z^2 . The straight lines buckle into curves — yet look at the crossings: every place two lines met at a right angle, the bent copies still meet at a right angle. Holding onto angles like that is what makes a map conformal . Straight lines became parabolas, but the right angles survived — that preserved angle is the whole idea of conformality. Drive the map: an angle that doubles, a corner that holds Drag the point z around the left plane and watch where w = z^2 drops it on the right. Its angle from the axis doubles ( ) and its distance from the origin squares ( |w| = |z|^2 ) . But notice the little right-angle corner riding on z : the map copies that corner onto w untouched. Distances and headings change; the angle between directions does not. Wherever , the corner stays a perfect — the map is conformal everywhere except the one point where f'(z) = 2z = 0 . Bend an entire half-plane into a disk

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