Complex Functions as Maps

Squaring, inverting and designing wings

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Complex functions are maps of the plane: z² squares lengths and doubles angles; 1/z turns the plane inside out.
• Such maps preserve angles (conformal), carrying physics solutions to new shapes.
• The Joukowski map turns circles into airfoils.

Steps

1. Multiply everything by i
2. Squaring and inverting
3. Squares stay square
4. Square it
5. Design a wing
6. Why angles matter
7. Turned inside out