Conformal Mapping

Holomorphic functions with nonzero derivative are exactly the angle-preserving (conformal) maps of the plane. This geometric side of complex analysis solves boundary-value problems in electrostatics and fluid flow by deforming a hard domain into an easy one, and its infinite-dimensional symmetry in two dimensions is the engine of 2D conformal field theory. This page builds on the Cauchy–Riemann picture of holomorphic.md.

Conformality

At a point where $f^{\prime }(z_0)\ne 0$, a holomorphic map acts to first order as multiplication by the complex number $f^{\prime }(z_0)$ — a rotation by $\arg f^{\prime }(z_0)$ and scaling by $|f^{\prime }(z_0)|$ (the CR Jacobian of holomorphic.md). Rotation and uniform scaling preserve angles between curves. Hence:

Holomorphic ⇔ conformal. A map is angle-preserving and orientation-preserving iff it is holomorphic with nonvanishing derivative. At critical points ($f^{\prime }(z_0)=0$) angles are multiplied by the order of the zero.

Because conformal maps preserve angles and take harmonic functions to harmonic functions (a holomorphic change of variables commutes with ${\nabla }^2$), they map one solution of Laplace's equation to another — the basis of the method below.

Möbius transformations

The Möbius (fractional linear) transformations $$w=\frac{az+b}{cz+d},ad-bc\ne 0,$$ are the conformal automorphisms of the Riemann sphere $\mathbb{C}\cup \{\infty \}$. They form a group $\cong PSL(2,\mathbb{C})$ — the same group that double-covers the Lorentz group (celestial- sphere action of Lorentz boosts). Key properties: they map circles-and-lines to circles-and-lines, are triply transitive (any three points to any three), and preserve the cross-ratio. Subclasses map the disc and half-plane to each other — the standard-domain toolkit.

The Riemann mapping theorem

Riemann mapping theorem. Any simply connected domain $U⊊\mathbb{C}$ (other than $\mathbb{C}$ itself) is conformally equivalent to the open unit disc.

So every reasonable simply connected region can be holomorphically straightened to the disc — an extraordinary uniformization. (For multiply connected or compact domains the classification is subtler; the uniformization theorem handles Riemann surfaces, sorting them into sphere/plane/disc by genus.) The boundary correspondence (Carathéodory) makes this practical for solving Dirichlet problems. The existence proof — via normal families and an extremal problem — is given in its own page, and the Riemann-surface generalization is the uniformization theorem.

Solving physics by mapping

Because harmonic functions pull back to harmonic functions, a 2D potential problem on a complicated domain is solved by conformally mapping it to a simple one (disc, half- plane, strip), solving there, and mapping back:

• electrostatics — potential around oddly shaped conductors (edges, corners);
• ideal fluid flow — flow past an aerofoil via the Joukowski map;
• steady heat conduction and membrane problems.

The complex potential $\Phi (z)=\varphi +i\psi$ packages the potential $\varphi$ and stream function $\psi$ as a single holomorphic function, with field lines and equipotentials the two orthogonal families $u,v=$ const.

Two-dimensional conformal field theory

In exactly two dimensions the conformal group is infinite-dimensional: every holomorphic map is locally conformal, so the symmetry algebra is the infinite Virasoro algebra rather than a finite Lie group. This vast symmetry makes 2D conformal field theory exactly solvable and underlies string worldsheets, critical phenomena, and the CFT page. The holomorphic/antiholomorphic factorization of 2D CFT correlators is the direct physical descendant of the holomorphic-function theory in this folder.

References

• Ahlfors, Complex Analysis, Ch. 6.
• Needham, Visual Complex Analysis, Ch. 3–5.
• Di Francesco, Mathieu & Sénéchal, Conformal Field Theory, Ch. 5–6.