Conformal Mappings

In this lesson you’ll learn what makes a map conformal, work with Möbius transformations and their circle-preserving property, meet the Riemann mapping theorem, and see how a change of shape solves boundary-value problems.

A map is conformal at $z_0$ if it preserves angles between curves through $z_0$, including their sense of rotation.

An analytic function is conformal exactly where $f'(z_0)\ne0$.

The reason is the local picture from lesson 5. Near $z_0$,

$$f(z) \approx f(z_0) + f'(z_0)(z-z_0)$$

so to first order the map is multiplication by the fixed complex number $f'(z_0)$: a rotation by $\arg f'(z_0)$ and a scaling by $|f'(z_0)|$. Both operations leave angles alone.

Where $f'(z_0)=0$ conformality fails. For $f(z)=z^2$ at the origin, angles are doubled; in general, a zero of order $m$ in $f'$ multiplies angles by $m+1$. Such points are called critical points, and a conformal map must avoid them.

Note the scaling factor $|f'|$ varies from point to point, so shapes are preserved only infinitesimally. A large square does not stay square; a tiny one does.

The most useful family:

$$w = \frac{az+b}{cz+d}, \qquad ad-bc\ne0$$

Their properties are worth knowing in full because they do most of the work in applications.

• Bijections of $\hat{\mathbb{C}}$. Every Möbius map is invertible on the Riemann sphere, with inverse $z = \frac{dw-b}{-cw+a}$.
• Group under composition, isomorphic to $PGL_2(\mathbb{C})$. Composing two gives another.
• Built from four elementary pieces: translation, rotation, scaling, and inversion. Every Möbius map factors into those.
• Conformal everywhere on $\hat{\mathbb{C}}$, since $w' = \frac{ad-bc}{(cz+d)^2}$ never vanishes.
• Circles go to circles, where a line counts as a circle through $\infty$. This is the single most useful property.
• Three points determine the map. Given three distinct source points and three distinct targets, exactly one Möbius transformation does the job.

That last fact is the practical construction tool: to map one region to another, pick three boundary points and their images, and the map is forced.

The figure shows the standard example,

$$w = \frac{z-i}{z+i}$$

which takes the upper half plane onto the unit disc. Checking with three points: $i\mapsto0$, $0\mapsto-1$, and $\infty\mapsto1$. Since the real axis is a “circle” through $\infty$ and it contains 0 and $\infty$, its image is a circle through $-1$ and $1$; a third real point pins it down as the unit circle.

The figure also verifies the containment numerically: the largest modulus among all sampled image points is safely under 1.

Riemann mapping theorem. Any simply connected domain in $\mathbb{C}$ that is not all of $\mathbb{C}$ can be mapped conformally and bijectively onto the unit disc.

This is one of the most remarkable existence theorems in mathematics. Any simply connected region - a square, a half plane, the interior of a fractal-looking curve, the complement of a slit - is conformally the same as a disc.

Two caveats keep it honest. It is an existence theorem: it gives no formula, and finding an explicit map for a given region can be very hard or practically impossible. And the excluded case $\mathbb{C}$ itself is genuinely excluded, since a bijection from $\mathbb{C}$ onto the disc would be a bounded entire function, hence constant by Liouville.

Multiply connected regions are not covered, and cannot be: an annulus is conformally equivalent only to annuli with the same ratio of radii, which is a genuine invariant.

Worth having memorized, since most problems reduce to composing these:

Map	Effect
$w = az+b$	translate, rotate, scale
$w = 1/z$	invert; exchanges inside and outside of the unit circle
$w = \frac{z-i}{z+i}$	upper half plane $\to$ unit disc
$w = z^2$	opens a quarter plane to a half plane
$w = e^z$	horizontal strip of height $\pi$ $\to$ upper half plane
$w = \operatorname{Log}z$	upper half plane $\to$ horizontal strip
$w = z+\frac1z$	circle $\to$ segment; the Joukowski map
$w = \frac{z-a}{1-\bar az}$	disc $\to$ disc, moving $a$ to the centre

The strategy for a hard region is to compose: get to a half plane, then to a disc.

Here is the payoff. If $u$ is harmonic - satisfying $\nabla^2u=0$ - and $f$ is conformal, then $u\circ f$ is harmonic too. Laplace’s equation is preserved by conformal maps.

So to solve a boundary-value problem on an awkward region $D$:

1. Find a conformal map $f$ from $D$ to the unit disc.
2. Transport the boundary data along $f$.
3. Solve on the disc, where the answer is a standard formula.
4. Pull the solution back through $f^{-1}$.

Since Laplace’s equation governs steady-state temperature, electrostatic potential, and incompressible irrotational flow, this single technique addresses all three. Before numerical methods it was the only way to handle non-trivial geometry, and it is still how the theory is organized.

Example 1: Where is $f(z)=z^2+z$ conformal?

Solution. $f'(z) = 2z+1$, which vanishes at $z=-\frac12$.

$$\text{conformal on } \mathbb{C}\setminus\{-\tfrac12\}$$

At $z=-\frac12$ the derivative has a simple zero, so angles there are doubled. ∎

Example 2: Map the upper half plane to the unit disc.

Solution. Use $w = \frac{z-i}{z+i}$. Verify with the three-point method:

$$i\mapsto\frac{0}{2i}=0, \qquad 0\mapsto\frac{-i}{i}=-1, \qquad \infty\mapsto1$$

The real axis passes through 0 and $\infty$, so its image is a circle through $-1$ and 1. Testing $z=1$ gives $w = \frac{1-i}{1+i} = -i$, which is on the unit circle, so the image circle is $|w|=1$.

Since $i$, an interior point, maps to the centre, the interior goes to the interior. ✓

Example 3: Map the first quadrant to the upper half plane.

Solution. The first quadrant is the sector $0<\arg z<\frac\pi2$. Squaring doubles arguments:

$$w = z^2$$

gives $0<\arg w<\pi$, the upper half plane. ∎

Composing with Example 2 sends the first quadrant to the disc via $w = \frac{z^2-i}{z^2+i}$. Composition is how every non-elementary region gets handled.

Example 4: Map the strip $0<\operatorname{Im}z<\pi$ to the unit disc.

Solution. Two steps. First $\zeta = e^z$ takes the strip to the upper half plane, from the mapping lesson. Then $w = \frac{\zeta-i}{\zeta+i}$ takes that to the disc.

$$w = \frac{e^z-i}{e^z+i}$$

∎

Example 5: Build the Möbius map sending $1\mapsto0$, $i\mapsto1$, $-1\mapsto\infty$.

Solution. Reading off the requirements: a zero at $z=1$ means a factor $(z-1)$ on top; a pole at $z=-1$ means $(z+1)$ on the bottom. So

$$w = k\cdot\frac{z-1}{z+1}$$

Fix $k$ using $i\mapsto1$:

$$1 = k\cdot\frac{i-1}{i+1}$$

Now $\frac{i-1}{i+1} = \frac{(i-1)(1-i)}{(1+i)(1-i)} = \frac{i-i^2-1+i}{2} = \frac{2i}{2}=i$, so $k\cdot i = 1$ and $k = -i$.

$$w = -i\cdot\frac{z-1}{z+1}$$

∎

Zeros and poles are read straight off the requirements, which makes these constructions quick once you see the pattern.

Example 6: Verify that a conformal map preserves harmonicity.

Solution. Let $u$ be harmonic on $D'$ and $f:D\to D'$ analytic. Locally $u = \operatorname{Re}F$ for some analytic $F$ on $D'$, since a harmonic function has a harmonic conjugate on a simply connected region.

Then $F\circ f$ is analytic on $D$ as a composition of analytic functions, and

$$u\circ f = \operatorname{Re}(F\circ f)$$

is the real part of an analytic function, hence harmonic. ∎

The proof needs no computation with second derivatives. Harmonic means “real part of analytic,” and analyticity is obviously preserved by composition.

Example 7: The Joukowski map.

Describe what $w = z+\frac1z$ does to the unit circle.

Solution. On $|z|=1$ write $z=e^{i\theta}$:

$$w = e^{i\theta}+e^{-i\theta} = 2\cos\theta$$

which is real and ranges over $[-2,2]$. So the unit circle collapses onto the segment $[-2,2]$, traversed twice.

The map has critical points where $w' = 1-\frac{1}{z^2} = 0$, that is $z=\pm1$, and those map to $w=\pm2$, the endpoints of the segment. The corners of the image are exactly the images of the critical points.

Applying the same map to a circle slightly offset from the unit circle produces a smooth curve with one sharp corner: an aerofoil. That is how the map earns its place in aerodynamics.

Aerofoil theory. The Joukowski map turns the solvable flow around a cylinder into the flow around a wing, and the resulting Kutta–Joukowski theorem gives lift in closed form. It is the historical foundation of aerodynamics.

Electrostatics and capacitance. Field configurations for awkward electrode shapes are computed by mapping to a parallel-plate or coaxial geometry. Handbook formulas for capacitance of odd-shaped conductors were obtained this way.

Heat sink and cooling design. Steady temperature distributions in fins and irregular cross-sections were computed by conformal mapping long before finite-element analysis, and the mapped solutions remain useful as benchmarks.

Mesh generation. Structured grids for computational fluid dynamics are often built by conformal maps, because the rotation-and-scaling property keeps cells well shaped and the numerics stable.

Map projections. The Mercator projection is conformal, which is why straight lines on it are constant-bearing courses. Stereographic projection is conformal too, and is used in crystallography for pole figures.

Medical imaging and brain mapping. Conformal flattening of curved cortical surfaces to a disc lets anatomical data from different subjects be compared in a common coordinate system, and the angle preservation is what keeps local structure recognizable.