Angle preservation
Definition. A holomorphic map \(f\) is conformal at \(a\) if \(f'(a)\ne0\). Its derivative is multiplication by the complex number \(f'(a)\), so infinitesimal lengths are scaled by \(|f'(a)|\) and angles are rotated by \(\arg f'(a)\).
Local inverse theorem
Theorem. If \(f\) is holomorphic near \(a\) and \(f'(a)\ne0\), then \(f\) is biholomorphic between neighborhoods of \(a\) and \(f(a)\). The inverse is holomorphic and satisfies \((f^{-1})'(f(a))=1/f'(a)\).
Critical points
Remark. If \(f'(a)=0\) and \(f(z)-f(a)=(z-a)^m g(z)\), \(m\ge2\), then locally the map multiplies angles by \(m\). Critical points are precisely where conformality fails.
Biholomorphic equivalence
Definition. Two domains are conformally equivalent if there is a bijective holomorphic map between them whose inverse is holomorphic. Such maps preserve analytic structure but not necessarily Euclidean size or shape.
Mapping boundaries
Example. The exponential map sends horizontal strips to annuli or punctured planes depending on the strip width. The map \(z\mapsto z^2\) sends a half-plane angle to a full-plane angle, illustrating angle multiplication at a critical point.
Geometry from analysis
Remark. Conformal mapping theory is where complex differentiability becomes geometry: analytic formulas classify and transform planar shapes.
Local conformality and global injectivity
A holomorphic map with nonzero derivative is locally angle preserving and a local diffeomorphism. A conformal equivalence between domains is a biholomorphic map. Local conformality does not imply global injectivity: \(e^z\) has no critical points but is \(2\pi i\)-periodic.
Proposition. If \(f:\Omega\to\Omega'\) is holomorphic and injective, then \(f'(z)\ne0\) throughout \(\Omega\), and its inverse is holomorphic with \[ (f^{-1})'(f(z))=\frac1{f'(z)}. \]
Proof. If \(f'\) vanished to order \(m-1\), then locally \(f(z)-f(a)=(z-a)^mg(z)\), \(m\ge2\), and nearby regular values would have \(m\) preimages, contradicting injectivity. The real inverse function theorem then gives a \(C^1\) inverse; its differential is multiplication by \(1/f'(z)\), so it satisfies Cauchy--Riemann.
Boundary correspondence and transformed geometry
Conformal maps preserve angles but not lengths or areas: \[ \mathrm{d}s_{\Omega'}=|f'(z)|\,\mathrm{d}s_\Omega, \qquad \mathrm{d}A_{\Omega'}=|f'(z)|^2\,\mathrm{d}A_\Omega. \] Harmonic functions pull back: if \(u\) is harmonic on \(\Omega'\), then \(u\circ f\) is harmonic on \(\Omega\). This lets one transfer Dirichlet problems and Green functions between conformally equivalent domains.
Standard transformations
- \(z\mapsto (z-i)/(z+i)\) maps the upper half-plane to the unit disc.
- \(z\mapsto z^\alpha\), with a chosen logarithm branch, maps sectors to sectors and can straighten corners.
- \(z\mapsto e^z\) maps horizontal strips to sectors or annuli and the full plane to the punctured plane as a covering map.
- Joukowski's map \(z+\frac1z\) maps the exterior of the unit disc to the complement of a line segment, with branching at \(\pm1\).
Conformal invariants
Simple connectedness, connectivity of complements, harmonic measure, and the hyperbolic metric are conformally invariant in their natural senses. Euclidean diameter and boundary smoothness are not. To disprove equivalence, one seeks an invariant; to construct a map, one composes standard maps while tracking domains and branch choices at every step.