Conformal mapping

Definition

Suppose $U$ is an open subset of $\mathbb{C}$. A map $f:U\to \mathbb{C}$ (or to a subset of $\mathbb{C}$ is termed a conformal mapping if it satisfies the following equivalent conditions:

• $f$ is a holomorphic function and $|f^{\prime }(z)|\ne 0$ for all $z\in U$
• $f$ maps smooth curves to smooth curves, and preserves both the magnitude and orientation of angle between curves at their intersection.

Relation with other properties

Weaker properties

• Isogonal mapping: This is a smooth mapping that preserves the magnitude of angles, but not necessarily the orientation. An isogonal mapping could be conformal or anticonformal