Conformal Maps and Geometric Function Theory

Part 9, Chapter 9: Complex Analysis, Holomorphic Functions

Learning objectives

Define conformal maps as holomorphic functions with nonzero derivative
Explain why $f'(z) \neq 0$ implies preservation of angles and orientation at $z$
Apply Mobius transformations to map standard domains (half-plane, disk, slit plane)
State the Riemann mapping theorem

Conformal maps are the angle-preserving transformations of the plane. They are exactly the holomorphic functions whose derivative does not vanish. The angle preservation comes directly from the Cauchy-Riemann equations: the derivative $f'(z_0)$ acts on tangent vectors as multiplication by a complex number, which is a rotation by $\arg f'(z_0)$ combined with a scaling by $|f'(z_0)|$ . Because complex multiplication is the same in every direction, the relative angles between curves through $z_0$ are preserved, even though shapes are stretched. This is why conformal maps are the natural language of 2D potential theory and aerodynamic shape design.

The conformal condition

A holomorphic function $f: U \to \mathbb{C}$ is conformal at $z_0 \in U$ if $f'(z_0) \neq 0$ . At such a point, two smooth curves crossing at $z_0$ with angle $\theta$ between their tangents map to two curves crossing at $f(z_0)$ with the same angle $\theta$ (and the same sense of rotation). Where $f'(z_0) = 0$ , angles are multiplied by the order of the zero: $f(z) = z^2$ doubles all angles at $0$ , $f(z) = z^3$ triples them, and so on.

Mobius transformations

The Mobius transformations (also called linear fractional transformations) are the maps

$T(z) = \frac{az + b}{cz + d}, \qquad ad - bc \neq 0.$

Each Mobius transformation is conformal everywhere on $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ (the Riemann sphere), and the collection of all Mobius transformations forms a group under composition (isomorphic to $PSL(2, \mathbb{C})$ ). Their key property: they map "circles or straight lines" to "circles or straight lines" (where straight lines are viewed as circles through infinity).

Three handy reference maps:

Upper half-plane to unit disk: $T(z) = (z - i)/(z + i)$ .

Unit disk automorphisms: for $|a| < 1$ and $|\theta| \leq \pi$ : $T(z) = e^{i\theta} (z - a)/(1 - \bar{a} z)$ .

Three-point pinning: there is exactly one Mobius transformation sending three given points to three other given points, found by solving 3 linear equations in $a, b, c, d$ .

The widget lets you watch a Mobius transformation in action. Try $T(z) = 1/z$ : the unit disk inverts to itself (mapping center to infinity and vice versa), the real axis maps to itself, and angles are preserved everywhere except at $z = 0$ and $z = \infty$ where the map is conformal-as-a-Riemann-sphere-map.

The Riemann mapping theorem

The Riemann mapping theorem is the central existence statement in geometric function theory: every simply connected open subset of $\mathbb{C}$ other than $\mathbb{C}$ itself is conformally equivalent to the open unit disk $\mathbb{D}$ . That is, for any such domain $U$ and any point $z_0 \in U$ , there exists a bijective conformal map $\varphi: U \to \mathbb{D}$ with $\varphi(z_0) = 0$ . The map is unique once you also specify $\arg \varphi'(z_0)$ .

The theorem is a pure existence statement; it does not provide an explicit formula. For specific domains (rectangles, ellipses, strips, polygons) explicit maps are known (Schwarz-Christoffel formula for polygons). For more exotic domains the map can be approximated numerically.

Where this shows up

Aerodynamics: The Joukowski transformation $w = z + 1/z$ maps a circle in the $z$ -plane to an airfoil-shaped curve in the $w$ -plane. Flow around a cylinder (easy) pulls back to flow around an airfoil (hard) via this conformal map. Early aircraft wings were designed exactly this way.

Cartography: The Mercator projection is a conformal map of (a portion of) the sphere to the plane. Loxodromes (constant-bearing paths) become straight lines on the chart, the property that made Mercator useful for navigation for four centuries despite the area distortion.

Electrostatics and heat conduction: Laplace's equation is invariant under conformal change of variables (in 2D). Engineers solve the equation on simple domains (half-plane, disk) and conformally pull the solution back to complicated ones (capacitor plate edges, fin geometries).

Medical imaging: Brain-surface parametrization for MRI registration uses conformal maps from the cortical surface to the sphere or disk. This is how neuroimaging software aligns brains across patients while preserving angular relationships of anatomical structures.

Computer graphics & texture mapping: Quasiconformal and conformal parametrizations of 3D surfaces produce distortion-minimal texture coordinates, the technical name for "draw a picture on this curved object without warping faces."

Pause and think: Why does $f(z) = z^2$ fail to be conformal at $z = 0$ even though it is entire? Compute $f'(0) = 0$ . Any small angle at the origin gets doubled by $f$ : two curves meeting at $90^\circ$ in the $z$ -plane meet at $180^\circ$ in the $w$ -plane, they become tangent! The vanishing derivative is the failure mode.

Try it

Verify that $T(z) = (z - i)/(z + i)$ maps the real axis to the unit circle. (Hint: for $x \in \mathbb{R}$ , compute $|T(x)|^2 = |x - i|^2 / |x + i|^2 = (x^2 + 1)/(x^2 + 1) = 1$ .)

Find a Mobius transformation that sends $0, 1, \infty$ to $1, i, -1$ respectively. (Hint: parametrize $T(z) = (az + b)/(cz + d)$ , plug in the three constraints, solve.)

Use the complex-plane widget to visualize $f(z) = e^z$ on the strip $\{x + iy : 0 < y < \pi\}$ . What domain in the $w$ -plane is the image? (Answer: the upper half-plane.)

True or false: by Riemann's mapping theorem, there exists a conformal bijection between the unit disk and the open first quadrant $\{z : \operatorname{Re}(z) > 0, \operatorname{Im}(z) > 0\}$ . (Answer: True, the quadrant is simply connected and not all of $\mathbb{C}$ .)

A trap to watch for

Conformality is a pointwise condition, but injectivity is a global condition. The map $f(z) = e^z$ is conformal at every point of $\mathbb{C}$ (its derivative $e^z$ never vanishes), yet $f$ is NOT injective on $\mathbb{C}$ , $e^{z + 2\pi i} = e^z$ . A conformal map can still wrap a domain around itself many times. The Riemann mapping theorem is more demanding: it gives a bijective conformal map, which requires the domain to be simply connected. Conformal-but-not-injective examples like $e^z$ on $\mathbb{C}$ are routine; they only become bijections when restricted to a fundamental strip $\{0 < \operatorname{Im}(z) < 2\pi\}$ .

What you now know

You can identify conformal maps by checking $f'(z) \neq 0$ , apply Mobius transformations to map between half-planes and disks, and understand the existential power and explicit limitations of Riemann's mapping theorem. This concludes Chapter 9. In Chapter 10 we leave geometry for foundations: countable and uncountable infinities, the paradoxes of naive set theory, the axiom of choice, and Godel's incompleteness theorems, the limits of what formal mathematics can prove.

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References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, §9.8.

Ahlfors, L. V. (1979). Complex Analysis (3rd ed.). McGraw-Hill, ch. 3 and 6.

Stein, E. M., Shakarchi, R. (2003). Complex Analysis. Princeton University Press, ch. 8 (Conformal Mappings).

Conway, J. B. (1978). Functions of One Complex Variable I (2nd ed.). Springer, ch. 7 (The Riemann Mapping Theorem).

Brown, J. W., Churchill, R. V. (2014). Complex Variables and Applications (9th ed.). McGraw-Hill, ch. 8-9.

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