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?