Conformal mappings

A function is conformal iff is analytic with on . If is bijective, it is a conformal equivalence.

Conformal maps preserve angles.
Let be a curve, with . The angle of the curve with direction is . Let be a conformal map and let be a new curve. The angle is now:

Note that is well defined because .
Now if we introduce a second curve , then we see it also gets rotated by and hence the angle between and is preserved.

The reverse is also true!

Examples

• maps rectangles to sectors of annuli
• maps sectors of annuli to rectangles
• will also map the upper half plane to the strip (or whatever the branch of is)
• Mobius map swaps the imaginary axis and the unit circle, while keeping the real line fixed (useful.)

Riemann Mapping Theorem

Any simply connected domain that is not the whole is conformally equivalent to the unit disk.