Definition:Curvilinear Coordinate System/Complex Plane

Definition
Let $u + i v = f \left({x + i y}\right)$ be a complex transformation.

Let $P = \left({x, y}\right)$ be a point in the complex plane.

Then $\left({u \left({x, y}\right), v \left({x, y}\right)}\right)$ are the curvilinear coordinates of $P$ under $f$.