Conformal Mapping

1. Analytic Functions Preserve Angles

A mapping $w = f(z)$ from the $z$-plane to the $w$-plane is called conformal at a point $z_0$ if it preserves the angle between any two curves passing through $z_0$, both in magnitude and in sense (orientation).

Theorem: If $f(z)$ is analytic at $z_0$ and $f'(z_0) \neq 0$, then $f$ is conformal at $z_0$.

Proof: Consider two smooth curves $\gamma_1(t)$ and $\gamma_2(t)$passing through $z_0$ at $t = 0$. The tangent vectors at $z_0$ are$\gamma_1'(0)$ and $\gamma_2'(0)$. Under the mapping $f$, the image curves are $w_k(t) = f(\gamma_k(t))$, and by the chain rule:

Writing the derivative in polar form, $f'(z_0) = |f'(z_0)| e^{i\alpha}$, we see that multiplication by $f'(z_0)$ scales each tangent vector by the same factor $|f'(z_0)|$ and rotates it by the same angle $\alpha = \arg f'(z_0)$. Therefore the angle between the two tangent vectors is preserved:

This proves that both the magnitude and the orientation of the angle are preserved. The condition$f'(z_0) \neq 0$ is essential: at a zero of the derivative, angles are multiplied by the order of the zero.

Local geometry: Near $z_0$, the conformal map behaves like a local rotation by $\arg f'(z_0)$ combined with a scaling by $|f'(z_0)|$. This means that infinitesimally small shapes are preserved (circles map to circles, squares to squares), though globally the map can distort shapes significantly.

Critical points: If $f'(z_0) = 0$ with $f(z) - f(z_0) \sim (z - z_0)^n$near $z_0$, then angles at $z_0$ are multiplied by $n$. For example, the map $w = z^2$ doubles angles at the origin, so two curves meeting at right angles ($\pi/2$) are mapped to curves meeting at angle $\pi$.

2. Mobius Transformations

The most important class of conformal maps is the set of Mobius transformations(also called linear fractional or bilinear transformations):

where $a, b, c, d \in \mathbb{C}$. The condition $ad - bc \neq 0$ ensures the map is non-degenerate (not a constant).

2.1 Properties of Mobius Transformations

Derivative: Computing $T'(z)$ by the quotient rule:

Since $ad - bc \neq 0$, we have $T'(z) \neq 0$ for all $z \neq -d/c$, confirming conformality everywhere except at the pole.

Composition: The composition of two Mobius transformations is again a Mobius transformation, and the associated matrices multiply:

This means the set of Mobius transformations forms a group isomorphic to$\text{PGL}(2, \mathbb{C}) = \text{GL}(2, \mathbb{C}) / \mathbb{C}^*$.

Inverse: The inverse of $T(z) = \frac{az+b}{cz+d}$ is:

2.2 Circle-Preserving Property

A fundamental property of Mobius transformations is that they map circles and lines(generalized circles on the Riemann sphere) to circles and lines. To prove this, note that every Mobius transformation can be decomposed into a sequence of simpler maps:

• Translation: $z \mapsto z + b$ (shifts circles)
• Rotation/scaling: $z \mapsto az$ (rotates and scales circles)
• Inversion: $z \mapsto 1/z$ (maps circles/lines to circles/lines)

Since each elementary operation preserves generalized circles, and any Mobius transformation$T(z) = \frac{az+b}{cz+d}$ can be written as a composition of these operations:

(translate by $d/c$, invert, scale by $\frac{bc-ad}{c^2}$, translate by $\frac{a}{c}$), the circle-preserving property follows.

2.3 Three-Point Determination

A Mobius transformation is uniquely determined by specifying the images of three distinct points. Given $z_1, z_2, z_3$ mapping to $w_1, w_2, w_3$, the transformation is determined implicitly by the cross-ratio:

This is an immensely practical formula. For example, the Mobius transformation mapping the upper half-plane $\text{Im}(z) > 0$ to the unit disk $|w| < 1$ with $z_0$mapping to the origin is:

3. The Joukowski Transform and Airfoil Theory

One of the most celebrated applications of conformal mapping in physics is the Joukowski transformation:

where $c > 0$ is a real parameter. This maps circles passing near the point $z = -c$into shapes resembling airfoil profiles. Specifically, a circle of radius $a > c$ centered at a point slightly offset from the origin produces an asymmetric airfoil with a sharp trailing edge.

Derivation of the critical points: The derivative is:

This vanishes at $z = \pm c$. At these critical points, the map is not conformal and angles are doubled. The points $w = \pm 2c$ on the real axis are the images of the critical points, forming the endpoints of the airfoil. The sharp trailing edge arises because two smooth curves on the circle meeting at an angle are mapped to curves meeting at twice that angle.

Parametric form: For a circle of radius $a$ centered at the origin,$z = ae^{i\theta}$, the image is:

When $a = c$, this reduces to $w = 2c\cos\theta$, a flat plate of length $4c$along the real axis. When $a > c$, it produces an ellipse with semi-axes$a + c^2/a$ and $a - c^2/a$.

4. Schwarz-Christoffel Mapping

The Schwarz-Christoffel mapping provides a systematic method for constructing conformal maps from the upper half-plane (or the unit disk) onto the interior of arbitrary polygonal regions. This is essential for solving Laplace's equation in domains with straight-line boundaries.

4.1 Derivation of the Schwarz-Christoffel Formula

Consider a polygon in the $w$-plane with vertices $w_1, w_2, \ldots, w_n$ and interior angles $\alpha_1\pi, \alpha_2\pi, \ldots, \alpha_n\pi$. We seek a conformal map$w = f(z)$ from the upper half of the $z$-plane onto this polygon, with the real axis mapping to the boundary.

Let $x_1 < x_2 < \cdots < x_n$ be the preimage points on the real axis. Along each edge of the polygon, $\arg(dw) = \arg(f'(z)\,dz)$ is constant (since the edge is a straight line). At each vertex $w_k$, the direction of the boundary turns by the exterior angle$(1 - \alpha_k)\pi$. Therefore $\arg(f'(z))$ must jump by $(1 - \alpha_k)\pi$as $z$ passes through $x_k$ on the real axis.

A function whose argument changes by $(\alpha_k - 1)\pi$ at $x_k$ is$(z - x_k)^{\alpha_k - 1}$, since:

As $z$ moves along the real axis past $x_k$, $\arg(z - x_k)$ changes from$\pi$ (approaching from the left) to $0$ (departing to the right), so the argument of the factor changes by $-(\alpha_k - 1)\pi = (1 - \alpha_k)\pi$, exactly the required exterior angle.

Taking the product over all vertices, the Schwarz-Christoffel formula for the derivative is:

where $A$ is a complex constant controlling rotation and scaling. Integrating:

The constants $A$ and $B$ account for translation, rotation, and scaling of the polygon. The constraint from the angle sum of a polygon is:

4.2 Example: Mapping to a Half-Strip

Consider the semi-infinite strip $\{w : 0 < \text{Re}(w) < \pi/2, \; \text{Im}(w) > 0\}$. This has vertices at $w_1 = 0$ and $w_2 = \pi/2$ with interior angles $\alpha_1 = \alpha_2 = 1/2$(right angles). Choosing $x_1 = -1, x_2 = 1$:

Integrating, $w = A \arcsin(z) + B$. Choosing boundary conditions to match the vertices gives$w = \arcsin(z)$, or equivalently $z = \sin(w)$.

5. Applications to 2D Electrostatics

In two-dimensional electrostatics, the electric potential $\Phi(x,y)$ satisfies Laplace's equation:

The key connection to complex analysis is that the real and imaginary parts of any analytic function are harmonic (satisfy Laplace's equation). If $F(z) = \Phi(x,y) + i\Psi(x,y)$ is the complex potential, then:

• $\Phi(x,y)$ = electric potential (equipotential lines are level curves of $\Phi$)
• $\Psi(x,y)$ = stream function (field lines are level curves of $\Psi$)
• The electric field is $\mathbf{E} = -\nabla\Phi$, which can be recovered from $F'(z) = -E_x + iE_y$

5.1 Invariance of Laplace's Equation

Theorem: If $\Phi(u,v)$ is harmonic in the $w$-plane and$w = f(z)$ is conformal, then $\Phi(u(x,y), v(x,y))$ is harmonic in the$z$-plane.

Proof: Let $w = u + iv = f(z)$ where $f$ is analytic. By the chain rule and the Cauchy-Riemann equations ($u_x = v_y$, $u_y = -v_x$):

This means we can solve Laplace's equation in a complicated geometry by mapping it conformally to a simple geometry, solving there, and transforming back.

5.2 Example: Potential Near a Conducting Edge

Consider a thin conducting plate along the positive $x$-axis held at potential $V = 0$, with the potential approaching $V_0$ at large distances in the upper half-plane. The conformal map $w = z^{1/2}$ transforms the upper half-plane slit along the positive $x$-axis into the first quadrant, where the solution is simply $\Phi = \frac{2V_0}{\pi}\arg(w)$. Transforming back:

Near the edge ($r \to 0$), the electric field $E = -\nabla\Phi \propto r^{-1/2}$diverges — this is the well-known edge effect in electrostatics.

6. Applications to 2D Fluid Flow

For two-dimensional, incompressible, irrotational (potential) flow, the velocity field$\mathbf{v} = (v_x, v_y)$ satisfies:

We can write $\mathbf{v} = \nabla\phi$ for a velocity potential $\phi$ satisfying$\nabla^2\phi = 0$. The complex velocity potential is$W(z) = \phi(x,y) + i\psi(x,y)$ where $\psi$ is the stream function. The complex velocity is:

6.1 Flow Around a Cylinder

Uniform flow past a circular cylinder of radius $a$ has the complex potential:

Adding circulation $\Gamma$ (the Kutta condition in aerodynamics):

The Joukowski transformation $\zeta = z + c^2/z$ then maps this flow onto flow past an airfoil, enabling the computation of lift via the Kutta-Joukowski theorem:

where $\rho$ is the fluid density and $L$ is the lift per unit span. This remarkable result connects a topological quantity (the circulation) to a measurable force, and follows directly from the residue theorem applied to the Blasius integral.

6.2 Source, Sink, and Vortex Flows

Elementary building blocks of complex potential theory include:

• Source/Sink of strength $m$ at $z_0$: $W(z) = \frac{m}{2\pi}\ln(z - z_0)$, giving radial flow
• Vortex of circulation $\Gamma$ at $z_0$: $W(z) = \frac{i\Gamma}{2\pi}\ln(z - z_0)$, giving circular flow
• Dipole of strength $\mu$ at $z_0$: $W(z) = \frac{\mu}{2\pi(z - z_0)}$

Superposition of these elementary solutions, combined with conformal mappings to handle complicated boundary geometries, forms the basis of classical potential flow theory.

7. The Riemann Mapping Theorem

The theoretical foundation for conformal mapping techniques is the Riemann Mapping Theorem:

This theorem guarantees that any simply connected domain (other than the whole plane) can be mapped conformally onto the unit disk. The practical challenge is finding the explicit form of the map for a given domain — the Schwarz-Christoffel formula solves this for polygonal domains.

Physical significance: The Riemann Mapping Theorem tells us that any two-dimensional potential problem on a simply connected domain can, in principle, be reduced to a problem on the unit disk. This provides an existence guarantee for solution methods based on conformal mapping.

Limitations: The theorem does not apply to multiply connected domains (e.g., annular regions). For doubly connected domains, the uniformization theorem guarantees a conformal map to an annulus $\{z : r < |z| < 1\}$ where $r$ is determined by the conformal modulus of the domain.

8. Common Conformal Maps Reference

Here are some of the most frequently used conformal maps in physics:

9. Python Simulation: Conformal Mapping

This simulation demonstrates key conformal mapping concepts numerically: verifying angle preservation, computing Joukowski airfoils, Mobius transformations, and the Schwarz-Christoffel mapping for a rectangle.