Conformal Mappings: Concepts, Formulas and Examples

A conformal mapping is an analytic function f with f′(z) ≠ 0 that preserves angles — both their size and their orientation — between curves. Zoom in close enough at any point and the map looks like a rotation combined with a uniform scaling: shapes keep their angles even as their sizes change.

This matters because conformal maps let you transplant problems from ugly domains to nice ones. Solve Laplace's equation (heat flow, electrostatics, fluid flow) on a simple disc or half-plane, then map the solution back to the original complicated shape — the angles that define the physics are preserved.

Key Concepts

• Conformality condition: f analytic with f′(z₀) ≠ 0 is conformal at z₀; near z₀ it acts as multiplication by f′(z₀).
• Local geometry: arg(f′(z₀)) is the rotation angle and |f′(z₀)| is the local scaling factor.
• Critical points: where f′(z₀) = 0, conformality breaks — angles get multiplied (w = z² doubles angles at 0).
• Riemann mapping theorem: any simply connected proper subdomain of the complex plane can be mapped conformally onto the unit disc.
• Boundary correspondence: under mild conditions, the conformal map extends continuously to the boundary, matching boundary points to boundary points.

Formulas

• Local linearisation: f(z) ≈ f(z₀) + f′(z₀)(z − z₀); the factor f′(z₀) rotates by arg(f′(z₀)) and scales by |f′(z₀)|.
• Power map: w = z² is conformal except at z = 0; it doubles all angles there.
• Exponential map: w = eᶻ maps the horizontal strip 0 < Im(z) < π conformally onto the upper half-plane Im(w) > 0.
• Joukowsky map: w = z + 1/z maps the exterior of the unit circle to the exterior of a slit, and deforms circles into airfoil shapes — the classical aerodynamics map.
• Assumptions: conformality needs analyticity plus non-vanishing derivative; global one-to-oneness needs separate verification.

Worked Example

Show that w = z² maps the first quadrant Q = {Re(z) > 0, Im(z) > 0} onto the upper half-plane H = {Im(w) > 0}.

1. Write z = re^{iθ} with r > 0 and 0 < θ < π/2 (the first quadrant in polar form).
2. Then w = z² = r²e^{i2θ}, with r² > 0 and 0 < 2θ < π.
3. An argument between 0 and π means Im(w) = r²sin(2θ) > 0, so w lies in the upper half-plane.
4. Conversely, any w with 0 < arg(w) < π has a square root with 0 < arg(z) < π/2, so the map is onto H.
5. Numeric check: z = 1 + i gives w = 2i (Im = 2 > 0 ✓); z = 2 + i gives w = 3 + 4i (Im = 4 > 0 ✓). The quadrant's corner angle 90° at z = 0 becomes 180° — angle doubling exactly where f′(0) = 0 breaks conformality.

Common Misconceptions

• "Conformal means globally one-to-one." No — conformality is local. w = z² is conformal at every z ≠ 0 yet maps both z and −z to the same w.
• "Conformal maps preserve sizes." They preserve angles, not sizes: the scaling factor |f′(z)| varies from point to point.
• "Every analytic function is conformal." Only where its derivative is non-zero; at critical points angles are multiplied by the order of vanishing.

Real-World Applications

• Aerodynamics: the Joukowsky map transforms flow around a cylinder into flow around an airfoil, the founding computation of wing theory.
• Electrostatics and heat flow: complicated conductor geometries are mapped to half-planes where potential problems are trivial.
• Cartography: the Mercator projection is conformal — it preserves compass bearings, which is why sailors used it.

A conformal map is angle-preserving complex differentiability in geometric form: locally a rotation plus a scaling. The power, exponential, and Joukowsky maps show how the right conformal change of variables turns hard domains into easy ones — which is why conformal mapping remains a core tool in fluids, fields, and geometry.