WaywardHouse

51  Conformal mapping

Angle-preserving transformations and applications to potential theory

51.1 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:

  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 \(f'(z_0)\). Writing \(f'(z_0) = |f'(z_0)|e^{i\varphi}\) with \(\varphi = \arg f'(z_0)\), this multiplication scales every tangent by \(|f'(z_0)|\) and rotates it by \(\varphi\). Both curves get the same rotation, 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. \(w = z^2\) at \(z = 0\): \(f'(0) = 0\), so the map is not conformal at the origin. The angle between any two curves through 0 is doubled. Away from 0, \(f'(z) = 2z \neq 0\) and the map is conformal.

51.2 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 form a group under composition and are the most important class of conformal maps.

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. A Möbius transformation is completely determined by specifying three distinct points and their images. Given pairs \((z_1 \mapsto w_1)\), \((z_2 \mapsto w_2)\), \((z_3 \mapsto w_3)\), the transformation is found by 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).

51.2.1 Fixed points

The fixed points of a Möbius transformation \(w = (az+b)/(cz+d)\) satisfy \(z = (az+b)/(cz+d)\), i.e., \(cz^2 + (d-a)z - b = 0\). A generic Möbius transformation has two fixed points (possibly coincident or at \(\infty\)).

51.3 Standard conformal maps

51.3.1 \(w = z^2\)

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 (since arg doubles from \((0,\pi)\) to \((0,2\pi)\)).

The first quadrant \(\{0 < \arg z < \pi/2\}\) maps to the upper half-plane \(\{0 < \arg w < \pi\}\).

51.3.2 \(w = e^z\)

Maps the horizontal strip \(\{0 < \text{Im}(z) < \pi\}\) to the upper half-plane \(\{\text{Im}(w) > 0\}\): the strip width \(\pi\) becomes the half-plane argument range \((0, \pi)\).

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

51.3.3 \(w = \log z\)

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

51.4 The Joukowski transformation

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

This is the foundational map of thin-aerofoil theory.

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 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.

51.5 Application to potential theory

The connection between conformal mapping and the Laplace equation makes it the principal analytic tool for two-dimensional field problems.

51.5.1 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.

51.5.2 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.

This means: solve Laplace’s equation in a simple domain (disk, half-plane, strip), map the domain to the complicated geometry using an appropriate conformal map, 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.

51.6 Exercises

51.6.1 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 \(|z| = 3\), \(\text{Im}(z) \geq 0\).

51.6.2 Exercise 2: Image of a strip under \(w = e^z\)

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

51.6.3 Exercise 3: Möbius transformation from three point pairs

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

51.6.4 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.

51.6.5 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 now have the full analytic toolkit: complex differentiation and the Cauchy-Riemann equations, contour integration and Cauchy’s theorem, Laurent series and residues, and conformal mapping. The methods 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.