Imagine two smooth curves in the complex plane: where . The angle between the curves is determined by the angle between the tangent vectors at . The angle between the vectors can be thought of as since the curves meet at
Now let be analytic at and let us focus on just a small part of the two cures in question: let . These will be smooth curves as well, albeit in the domain space.
We have the chain rule and so and
Now look at the arguments: and PROVIDED Now calculate which is the angle between the original curves.
Any function that preserves angles in this manner is called conformal. So we showed that, if is analytic on an open disk and is not zero on the disk, then is conformal. Furthermore, our previous work shows that if is analytic and one-to-one on a disk, never zero on the disk. So functions which are analytic and one-to-one (at least locally) are conformal (at least locally).
Examples: find where are conformal. What about ?
Related exercise: where is analytic and one to one?