55  Conformal mapping

Angle-preserving transformations and applications to potential theory

55.1 What this chapter helps you do

Symbols to keep handy

These are the bits of notation you'll see a lot. If a line of symbols feels like a fence, read it out loud once, then keep going.

  • w = z + 1/z: Joukowski transformation

  • w = : Möbius transformation

Definitions to keep handy

These are the words we keep coming back to. If one feels slippery, come back here and steady it before you push on.

  • conformal map: A transformation that preserves angles (locally) even if it distorts sizes.

  • Mobius transformation: A fractional linear map (az+b)/(cz+d) that sends lines/circles to lines/circles.

  • Joukowski transform: A classic map that turns circles into airfoil-like shapes.

  • complex potential: A compact way to package potential and streamfunction together as one complex function.

Here is the main move this chapter is making, in plain terms. You do not need to be fast. You just need to keep the thread.

  • Coming in: An analytic function with non-zero derivative maps infinitesimally small shapes to similar shapes — it preserves angles. This conformal property means that a problem on a complicated domain can be transformed to a simple domain (a half-plane, a disk, a strip) where the solution is known, and then mapped back.

  • Leaving with: If f is analytic and f′(z₀) ≠ 0, then f is conformal at z₀: it preserves angles between curves and maps small circles to small circles. The Möbius transformation w = (az+b)/(cz+d) is the most important class — it maps circles and lines to circles and lines, and three points determine it uniquely. The Joukowski transform w = z + 1/z maps circles near the unit circle to aerofoil shapes. In potential theory, a conformal map w = f(z) transforms the complex potential F(z) = Φ + iΨ: if F(w) solves Laplace’s equation in the w-domain, then F(f(z)) solves it in the z-domain.

55.2 Conformal mappings

An analytic function w = f(z) is conformal at z_0 if f'(z_0) \neq 0. At such a point, the mapping:

Conformal map, in words

Near z_0, an analytic map with f'(z_0)\neq 0 behaves like a very simple geometric move: it rotates and scales small shapes.

So it can distort sizes and positions, but it does not shear angles.

  1. Preserves angles between curves (in both magnitude and orientation).
  2. Scales uniformly: all infinitesimal lengths at z_0 are multiplied by |f'(z_0)|.
  3. Rotates uniformly: all directions at z_0 are rotated by \arg f'(z_0).

Why angles are preserved. If two curves \gamma_1 and \gamma_2 meet at z_0 at angle \alpha, their images under f meet at f(z_0) at the same angle. The chain rule gives the tangent direction of the image curve as f'(z_0)\cdot\gamma'(t_0) — the original tangent vector multiplied by f'(z_0). Write f'(z_0) = |f'(z_0)|e^{i\varphi} where \varphi = \arg f'(z_0). Multiplying a complex number by e^{i\varphi} rotates it by \varphi and multiplying by |f'(z_0)| scales it. Both curves get the same rotation by \varphi, so the angle between them is unchanged.

At a critical point where f'(z_0) = 0, conformality fails: angles are multiplied by the order of the zero.

Example. For w = z^2 at z = 0: f'(0) = 0, so the map is not conformal there. The angle between any two curves through the origin is doubled. Away from the origin, f'(z) = 2z \neq 0 and the map is conformal everywhere else.

55.3 Möbius transformations

The Möbius transformation (or fractional linear transformation) is:

w = \frac{az + b}{cz + d}, \qquad ad - bc \neq 0

The condition ad - bc \neq 0 ensures that w is not constant. These transformations compose cleanly — applying one after another gives another Möbius transformation — which is why they are the most useful class of conformal maps in practice.

Key property: circles and lines map to circles and lines. In the extended complex plane — the ordinary plane with a single point at infinity added — a straight line can be viewed as a circle that passes through that point at infinity. Under a Möbius transformation, circles (including lines viewed this way) always map to circles (including lines).

Three-point determination. Specify three distinct points and their images and the transformation is fixed. Given pairs (z_1 \mapsto w_1), (z_2 \mapsto w_2), (z_3 \mapsto w_3), the transformation comes from the cross-ratio equation:

\frac{(w - w_1)(w_2 - w_3)}{(w - w_3)(w_2 - w_1)} = \frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}

The right side is a known number for each z. Cross-multiply to get a linear equation in w, then solve — the result is the unique Möbius transformation. Exercise 3 works through this procedure.

Standard mappings.

  • w = \dfrac{z - i}{z + i}: maps the upper half-plane \text{Im}(z) > 0 to the unit disk |w| < 1, and the real axis to the unit circle.

  • w = \dfrac{1+z}{1-z}: maps |z| < 1 (unit disk) to \text{Re}(w) > 0 (right half-plane).

55.3.1 Fixed points

A fixed point is a point that the transformation sends to itself. For w = (az+b)/(cz+d), setting z = w gives cz^2 + (d-a)z - b = 0. A generic Möbius transformation has two fixed points (possibly coincident or at \infty). Knowing the fixed points characterises the geometric type of the transformation — rotation, dilation, translation, or a combination.

55.4 Standard conformal maps

55.4.1 w = z^2

Squaring doubles the argument and squares the modulus. The upper half-plane \{z : \text{Im}(z) > 0\} maps to the full complex plane minus the non-negative real axis: argument ranges from (0, \pi) in the z-plane, doubling to (0, 2\pi) in the w-plane.

A more useful sub-case: the first quadrant \{0 < \arg z < \pi/2\} maps to the upper half-plane \{0 < \arg w < \pi\}. Doubling a right angle gives a straight angle.

55.4.2 w = e^z

Write z = x + iy. Then w = e^x e^{iy}, so |w| = e^x (any positive real value) and \arg w = y. The horizontal strip \{0 < \text{Im}(z) < \pi\} maps to the upper half-plane \{\text{Im}(w) > 0\}: the imaginary part of z becomes the argument of w.

Check the boundaries directly. The lower edge y = 0 gives e^{x+0i} = e^x > 0, which lands on the positive real axis (\arg w = 0). The upper edge y = \pi gives e^{x+i\pi} = -e^x < 0, which lands on the negative real axis (\arg w = \pi). The interior maps to the open upper half-plane.

More generally, the strip \{a < \text{Im}(z) < b\} maps to the sector \{a < \arg w < b\}.

55.4.3 w = \log z

The inverse of w = e^z: it maps the upper half-plane to the horizontal strip \{0 < \text{Im}(w) < \pi\}.

55.5 The Joukowski transformation

w = z + \frac{1}{z}

This map sits at the foundation of thin-aerofoil theory. The behaviour depends on which circle in the z-plane you start with.

The unit circle maps to a degenerate shape. For z = e^{i\theta}: w = e^{i\theta} + e^{-i\theta} = 2\cos\theta \in [-2, 2]

The unit circle collapses to the real interval [-2, 2] — a flat plate.

Circles near the unit circle map to aerofoil-like shapes. For z = re^{i\theta} with r > 1: w = \left(r + \frac{1}{r}\right)\cos\theta + i\left(r - \frac{1}{r}\right)\sin\theta

This is an ellipse with semi-axes a = r + 1/r and b = r - 1/r. As r \to 1^+, the ellipse degenerates toward the flat plate.

Off-centre circles. Shifting the circle in the z-plane before applying the Joukowski map breaks the ellipse’s symmetry and produces a curved aerofoil profile with a sharp trailing edge. This is the Joukowski aerofoil — the basis of early wing design.

Why it works for aerodynamics. The complex potential for incompressible irrotational flow around a cylinder of radius r is known: F(z) = U(z + r^2/z) for uniform flow at speed U. Composing with the Joukowski map gives the complex potential for flow around the aerofoil. The Kutta–Joukowski theorem then gives the lift per unit span as L = \rho U \Gamma, where \Gamma = \oint \mathbf{v}\cdot d\mathbf{s} is the circulation — the line integral of the tangential velocity component around the aerofoil.

55.6 Application to potential theory

Conformal mapping is the main analytic tool for two-dimensional field problems — fluid flow, electrostatics, heat conduction — because it transforms the Laplace equation in a complicated domain into the Laplace equation in a simple one, and solutions to Laplace’s equation compose under analytic maps.

55.6.1 Complex potential

For a two-dimensional incompressible irrotational flow (or electrostatic field, or heat conduction problem), the complex potential F(z) = \Phi(x,y) + i\Psi(x,y) has: - \Phi: velocity potential (or electrostatic potential, or temperature) - \Psi: stream function (or flux lines)

The complex velocity is dF/dz = v_x - iv_y, where (v_x, v_y) is the physical velocity vector. The sign convention (v_x - iv_y rather than v_x + iv_y) is standard in potential flow and follows from F = \Phi + i\Psi with v_x = \Phi_x, v_y = \Phi_y.

Since F is analytic, both \Phi and \Psi satisfy Laplace’s equation, and they are harmonic conjugates.

55.6.2 Transforming the domain

If w = f(z) is a conformal map, and F(w) is the complex potential in the w-domain, then F(f(z)) is the complex potential in the z-domain.

In practice: solve Laplace’s equation in a simple domain (disk, half-plane, strip), choose a conformal map from that domain to the complicated geometry, and the composition gives the solution in the complicated domain.

Example. Take G(w) = w (uniform flow in the w-plane) and the conformal map f(z) = z^2. The composed potential in the z-plane is G(f(z)) = z^2. That is, substitute w = z^2 directly: the result is a new function of z, not G evaluated at z. With f(z) = z^2, the z-plane flow has streamlines \Psi = \text{Im}(z^2) = 2xy = \text{const} — the rectangular hyperbola pattern of flow in a 90° corner.

55.7 Exercises

55.7.1 Exercise 1: Image of a circle under w = z^2

Find the image of the circle |z| = 3 under w = z^2. Then find the image of the half-circle |z| = 3, \text{Im}(z) \geq 0.

55.7.2 Exercise 2: Image of a strip under w = e^z

Find the image of the horizontal strip \{0 < \text{Im}(z) < \pi/2\} under w = e^z.

55.7.3 Exercise 3: Möbius transformation from three point pairs

Find the Möbius transformation mapping z_1 = 0 \mapsto w_1 = i, z_2 = 1 \mapsto w_2 = 0, z_3 = -1 \mapsto w_3 = \infty.

55.7.4 Exercise 4: Joukowski map — circle to ellipse

Apply w = z + 1/z to the circle |z| = 2. Find the semi-axes of the resulting ellipse.

55.7.5 Exercise 5: Potential theory — flow in a corner

The complex potential F(w) = w represents uniform horizontal flow in the w-plane. Apply the conformal map w = z^2 to find the complex potential for flow in the first quadrant (x > 0, y > 0) of the z-plane.

With these four chapters you have the core analytic methods: complex differentiation and the Cauchy-Riemann equations, contour integration and Cauchy’s theorem, Laurent series and residues, and conformal mapping. They connect directly to the rest of engineering mathematics — the Laplace equation in any two-dimensional domain, potential flow around aerofoils, inverse Laplace transforms via residues, and stability analysis in the complex frequency plane.