FKM Kompass
L3

Conformal mapping

A conformal map is a function of a complex variable that preserves angles between intersecting curves at every point where its derivative is nonzero — a consequence of holomorphy, not an extra assumption — which makes such maps the tool of choice for warping an awkward geometric region into a simple one (a disk, a half-plane) while keeping the local angular structure that a physical field's boundary conditions depend on.

Practise this concept

Every holomorphic function with a nonzero derivative at a point is, at that point, locally just a rotation and a uniform scaling — that's what having a single complex derivative actually means geometrically, since multiplying by a complex number f′(z₀) rotates by its argument and scales by its modulus, the same amount in every direction. A rotation-plus-uniform-scaling can distort distances freely but can never change the angle between two lines crossing at that point, which is exactly the definition of conformal: angle-preserving.

This breaks down at any point where f′(z) = 0, however holomorphic f still is there — the local behaviour is no longer a simple rotation-and-scale, and angles genuinely do distort. The classic example is w = z², which doubles angles at the origin (where the derivative vanishes) while still preserving them everywhere else.

The practical payoff is solving boundary value problems on awkward regions. Laplace's equation is conformally invariant: if u(x, y) is harmonic on a region and w = f(z) is a conformal map to a nicer region (a disk, a half-plane, a strip), then u composed with f⁻¹ is harmonic on the new region too. So the standard trick is to map an ugly domain to a standard one, solve the easy problem there (often by inspection or a known formula), and map the solution back — the angle preservation guarantees the boundary conditions carry over correctly, since the geometry that determined them locally hasn't been twisted, only reshaped.

Key ideas

Requires: Functions of a complex variable

Common misconceptions

  • Believes a conformal map preserves distances and shapes, like a rigid rotation, rather than only local angles — a conformal map can stretch a region enormously, it just can't twist the angle at which two curves cross.
  • Assumes every holomorphic function is conformal at every point of its domain, missing that angle preservation fails at any point where the derivative vanishes, even though the function is still perfectly holomorphic there.
  • Thinks mapping a complicated region to a disk or half-plane is only a geometric convenience with no bearing on solving a physical problem, missing that Laplace's equation is itself preserved under a conformal change of variables — so a solution on the simple region translates directly back into a solution on the original one.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.