conformal map

A conformal map is a transformation of the plane that preserves angles: wherever two curves cross at some angle, their images cross at the very same angle, with the same sense of rotation. It may stretch, shrink, bend and curl regions dramatically, but it never shears — infinitesimally small shapes are rotated and scaled but not distorted, so tiny squares map to tiny squares (not parallelograms) and tiny circles to tiny circles. This is the geometric face of holomorphicity.

Precisely, a holomorphic function f is conformal at a point z0 exactly when its derivative f'(z0) is nonzero. The reason is that, to first order, f acts near z0 like multiplication by the complex number f'(z0), and multiplication by a nonzero complex number is a rotation (by the argument of f'(z0)) combined with a scaling (by its modulus) — and rotation-plus-scaling preserves angles. A conformal map between two domains is a holomorphic bijection with nonzero derivative throughout, also called a biholomorphism.

Conformal maps are central to applications, transplanting solutions of physical problems (heat flow, electrostatics, fluid flow) from awkward regions to simple ones like a disc or half-plane. The deep Riemann mapping theorem says any simply connected open subset of the plane that is not all of C can be conformally mapped onto the unit disc. A caveat about the derivative condition: at a point where f'(z0) = 0 angles are not preserved but multiplied (a point where f'(z) = 0 of order k multiplies angles by k+1), so such points are genuinely non-conformal.