A complex-differentiable function in a plane region is said to be holomorphic.

Geometrically, holomorphic mappings are angle-preserving (except where the derivative is zero) because complex multiplication effects scaling and rotation. Under a holomorphic mapping with non-vanishing derivative, a fine grid of squares maps to a grid of near-squares.

Analytically, a holomorphic function satisfies the mean-value property: Its average value over an arbitrary disk is the value at the center of that disk.

Further, a holomorphic function is represented near an arbitrary point by a convergent power series. Consequently, a holomorphic function is rigid: uniquely determined throughout a region by its values in an arbitrarily small disk. Unlike a real-differentiable function, a holomorphic function cannot be modified in one place without changing its values in every disk. A holomorphic function is more analogous to a flexible web of linkages than to a single curve of wire.