The Exponential Map

Complex Analysis · Axiom Academy

How w = e^z bends the flat grid of the plane into rings and rays — a conformal map you can watch. 1. A Horizontal Line Becomes a Ray Fix a height and slide right. Every point on the horizontal line shares the same y = c , so every image shares the same angle — they all land on one ray from the origin. As x grows, the modulus |w| = e^x grows, so the image races outward along that fixed ray. A horizontal line in the z -plane (height c ) maps to a ray at angle c in the w -plane 2. A Vertical Line Becomes a Circle Now fix a column and slide up. Every point on the vertical line shares the same x = c , so every image has the same modulus |w| = e^c — they all sit on one circle centered at the origin. As y climbs, the angle winds around, so the image travels around that fixed circle, completing a full loop every time y advances by . Every point shares x = c , so |w| = e^c is the same for all of them — one circle. Climbing the line raises y , and carries the image around the circle. Advancing y by returns to the same point: . Shift the line right and the circle grows; the radius e^c is set entirely by c . Because of that loop, e^z is periodic Since one full turn of the circle costs exactly in y , the map repeats with period . So e^z is not one-to-one on all of — to get a bijection you restrict to a single horizontal strip of height . 3. The Grid Goes Polar — and Angles Survive

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