One has for a static charge distribution, ρ(r), that the electric field, E = ∇ϕ, satisfies \[ \nabla\cdot{\bf E} = \rho /\epsilon_0 , \] where

In regions devoid of charge, this equation yields the Laplace equation ∇²ϕ = 0.

Solutions to Laplace’s equation in 2D wedge domains have powerful applications in electrostatics and fluid flow, especially for modeling singularities, boundary layers, and field behavior near corners or edges. These problems arise naturally in engineering, physics, and applied mathematics.

Electrostatics Applications:

1. Electric Field Near Conducting Corners A wedge domain models the region near the corner of a conducting surface. Solving Laplace’s equation with Dirichlet conditions (fixed potential) or Neumann conditions (specified flux) reveals how the electric potential behaves near edges. For wedge angles α > π, the solution exhibits singular behavior, mimicking field intensification at sharp tips — crucial for understanding corona discharge and dielectric breakdown.

2. Capacitor Edge Effects In parallel plate capacitors with finite edges, the field near the edge resembles a wedge. The wedge solution helps quantify fringing fields, which affect capacitance and energy storage.

3. Electrostatic Shielding and Conformal Mapping Wedge domains are used in conformal mapping techniques to design electrostatic shields and solve problems involving complex geometries. Mapping wedge domains to simpler ones (e.g., half-plane) allows analytic solutions for charge distributions and potential contours.

Electrostatics Applications

1. Field Intensification at Conducting Corners

• A wedge models the region near a sharp conducting corner.
• Solving Laplace’s equation with Dirichlet conditions (fixed potential) shows how the electric field behaves near the tip.
• If the wedge angle α > π, the solution becomes singular near the origin: \[ u(r, \theta) \sim r^{\pi/\alpha} \sin\left(\frac{\pi \theta}{\alpha}\right) \]
• The electric field magnitude \( \displaystyle \quad |\nabla u| \sim r^{\pi/\alpha - 1} \quad \) blows up as r → 0.

2. Fringing Fields in Capacitors

• Near the edge of a parallel plate capacitor, the field lines bend outward.
• The wedge solution helps quantify fringing effects, which influence capacitance and energy storage.
• The asymptotic behavior near the origin (or corner) is critical for electrostatics: it determines field strength and energy density

Sources and Further Reading:

Laplace Equation: Theory and Applications
Laplace Equation in Physics and Engineering
Analytical Solution via Separation of Variables

Similarly, we can derive Laplace’s equation for an incompressible, ∇·v = 0, irrotational, ∇×v = 0, fluid flow. From well-known vector identities, we know that ∇ × ∇ϕ = 0 for a scalar function ϕ. Therefore, we can introduce a velocity potential, ϕ, such that v = ∇ϕ. Thus, ∇·v = 0 implies ∇²ϕ = 0. So, the velocity potential satisfies Laplace’s equation.

Fluid flow is probably the simplest and most interesting application of complex variable techniques for solving Laplace’s equation. So, we will spend some time discussing how conformal mappings have been used to study two-dimensional ideal fluid flow, leading to the study of airfoil design.

The study of fluid flow and conformal mappings dates back to Euler, Riemann, and many others. The method was further elaborated upon by physicists like Lord Rayleigh (1877) and applications to airfoil theory we presented in papers by Kutta (1902) and Joukowski (1906) on later to be improved upon by others.

Fluid Flow Applications

1. Stagnation Points and Corner Flow:
In potential flow theory, Laplace’s equation governs the velocity potential. Wedge domains model flow near sharp corners, such as in ducts, nozzles, or around obstacles. Solutions reveal stagnation zones, vortex formation, and pressure singularities.

2. Flow Past Slits and Cracks:
In 2D, flow past a slit or crack is modeled by a wedge with angle α = 2π. The solution describes streamlines and velocity fields, useful in aerodynamics, hydraulics, and microfluidics.

3. Lubrication and Thin Film Flow Wedge-shaped gaps arise in lubrication theory, where fluid flows between surfaces with angular separation. • Laplace solutions help predict pressure distribution, film thickness, and load capacity.

• The asymptotic behavior near the origin (or corner) is critical:
• For fluid flow: it governs velocity gradients and shear stress
• These solutions are foundational in fracture mechanics, MEMS design, antenna modeling, and boundary layer theory

Wedge-shaped gaps arise in lubrication theory, where fluid flows between surfaces with angular separation. Laplace solutions help predict pressure distribution, film thickness, and load capacity.    ■

Its derivation can be obtained via the method of Fourier transforms or conformal mapping from the unit disk to the half-plane.

Recall some properties and interpretation. The kernel \[ P(x - \xi, y) = \frac{1}{\pi} \cdot \frac{y}{(x - \xi)^2 + y^2} \] is the Poisson kernel, a fundamental solution that acts like a “smeared delta function” as y → 0. As y → 0, u(x, y) → f(x) in the sense of boundary limits (pointwise or in 𝔏^p depending on f).

Example: Let’s take \( \displaystyle \quad f(x) = \frac{1}{1 + x^2}. \quad \) Then \[ u(x, y) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{y}{(x - \xi)^2 + y^2} \cdot \frac{1}{1 + \xi^2} \, d\xi \] This integral can be evaluated numerically or symbolically (e.g., via residue calculus or convolution with known transforms).

We use separation of variables in Cartesian coordinates. Assume: \[ u(x, y) = X(x) Y(y) \] Substituting into Laplace’s equation, we obtain \[ X''(x) Y(y) + X(x) Y''(y) = 0 \quad \Rightarrow \quad \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = -\lambda \] This gives two ODEs: \[ \begin{split} X'' + \lambda X &= 0 , \\ Y'' - \lambda Y &= 0 . \end{split} \] General Solution \[ u(x, y) = \int_0^\infty \left[ A(\lambda) \sin(\sqrt{\lambda} x) + B(\lambda) \cos(\sqrt{\lambda} x) \right] \cdot \left[ C(\lambda) e^{-\sqrt{\lambda} y} + D(\lambda) e^{\sqrt{\lambda} y} \right] \, {\text d}\lambda \] To satisfy decay and boundary conditions, we choose appropriate terms and determine coefficients via Fourier sine/cosine transforms.

Let us use the Conformal Mapping Approach. Alternatively, map the quarter-plane to the upper half-plane using: \[ w = z^2, \quad \text{where } z = x + {\bf j}\,y \in \mathbb{C} . \] We have \[ z = x + {\bf j}\,y \qquad \Longrightarrow \qquad w = x^2 - y^2 + 2{\bf j}xy . \] The boundaries become \[ \begin{split} x > 0,\ y = 0 \qquad \Longrightarrow \qquad w \in \mathbb{R}_+ , \\ x = 0, \ y > 0 \qquad \Longrightarrow \qquad w \in \mathbb{R} . \end{split} \] So the positive real and imaginary axes in z-space map to the positive and negative real axes in w-space. This maps the quarter-plane x > 0, y > 0 to the upper half-plane Im(w) > 0.

Let U(w) be harmonic in ℍ with boundary data: \[ \begin{split} U(\xi) &= f(\sqrt{\xi}) \quad \mbox{for}\quad \xi > 0 , \\ U(\xi) &= g(i\sqrt{-\xi}) \quad \mbox{for} \quad \xi < 0 . \end{split} \] where F(ξ) is the combined boundary data: \[ F(\xi) = \begin{cases} f(\sqrt{\xi}), & \xi > 0 , \\ g(i\sqrt{-\xi}), & \xi < 0 . \end{cases} \] Then apply the Poisson integral formula in w-coordinates: \[ U(w) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\text{Im}(w) \cdot F(\xi)}{(\text{Re}(w) - \xi)^2 + (\text{Im}(w))^2} \, {\text d}\xi , \] Pull Back to the Quarter-Plane Finally, define: \[ u(z) = U(z^2) . \] This gives a harmonic function in the quarter-plane that satisfies the original boundary conditions. \[ u(z) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\text{Im}(w) \cdot F(\xi)}{(\text{Re}(w) - \xi)^2 + (\text{Im}(w))^2} \, {\text d}\xi , \] where F(ξ) is the transformed boundary data.

Example: Let’s take: \[ \begin{split} f(x) &= \sin(ax) , \\ g(y) &= 0 \end{split} \] Then: \[ \begin{split} F(\xi) &= \sin(a\sqrt{\xi})\quad \mbox{for} \quad \xi > 0 , \\ F(\xi) &= 0 \quad \mbox{for}\quad \xi < 0 . \end{split} \] So: \[ U(w) = \frac{1}{\pi} \int_0^{\infty} \frac{\text{Im}(w) \cdot \sin(a\sqrt{\xi})}{(\text{Re}(w) - \xi)^2 + (\text{Im}(w))^2} \, d\xi \] and the solution in the quarter-plane is: \[ u(z) = U(z^2) \] So the solution becomes: \[ u(x, y) = \sin(ax) \cdot e^{-a y} \]

This solution is constructed from the real part of the analytic function: \[ f(z) = z^2 + z^4 = r^2 e^{2{\bf j}\theta} + r^4 e^{4{\bf j}\theta} . \] Taking the real part gives: \[ u(r, \theta) = r^2 \cos(2\theta) + r^4 \cos(4\theta) . \] It satisfies Laplace’s equation in the wedge. It obeys homogeneous Neumann conditions on both boundaries (θ = 0 and θ = π/2) because the angular derivatives vanish there.    ■

Let z = x + ⅉy = r e^ⅉθ be the complex variable with ⅉ being the imaginary unit so ⅉ² = −1. Some times, we will denote this unit vector of complex plane ℂ as j; note that mathematicians prefer to use Euler's notation i for this unit. The wedge domain corresponds to: \[ \arg(z) \in (0, \alpha) . \] We map this wedge to the upper half-plane using: \[ w = z^{\pi/\alpha} . \] This maps the wedge arg(z) ∈ (0, α) to arg(w) ∈ (0, π), i.e., the upper half-plane.

Now we solve in the Upper Half-Plane problem. In the w-plane, we seek an analytic function U(w) such that u = ReU(z^π/α)exp(ⅉπθ/α). Let U(w) be harmonic in the upper half-plane with boundary values prescribed on the real axis. The Dirichlet problem in the upper half-plane can be solved using the Poisson integral formula: \[ U(x + {\bf j}y) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{y\, g(t)}{(x - t)^2 + y^2} {\text d}t , \] where g(t) is the boundary data on the real axis and j is the imaginary unit (also denoted by ⅉ), so j² = −1.

To use this formular, we need to express the boundary data f₁(r) and f₂(r) in terms of the real axis of the w-plane. That is, we define: \[ g(w) = \begin{cases} f_1(r), & \text{if } w = r^{\pi/\alpha} \text{ for } \theta = 0 \\ f_2(r), & \text{if } w = r^{\pi/\alpha} e^{i\pi} \text{ for } \theta = \alpha \end{cases} . \] Finally, we pull back the solution. Once U(w) is found, the solution in the original wedge domain is: \[ u(z) = \Re[U(z^{\pi/\alpha}) . \] This method dates back to Riemann and Schwarz, and was popularized in the context of electrostatics and fluid flow. The wedge geometry is particularly amenable to complex variable techniques due to the power mapping z^{\pi/\alpha}, which linearizes the angular domain

Connection to separation of variables:    Both methods, Analytic Functions and Eigenfunction Expansions, solve the same PDE — Laplace’s equation — but they approach it from different angles:

• Complex variable method: uses conformal mapping and analytic functions, exploiting the fact that harmonic functions are real parts of holomorphic functions.
• Separation of variables: uses orthogonal eigenfunction expansions in polar coordinates, typically involving sine and cosine functions in θ and powers of r.

The general solution is: \[ u(r, \theta) = \sum_{n=1}^\infty A_n r^{n\pi/\alpha} \sin\left( \frac{n\pi \theta}{\alpha} \right) . \] This function satisfies the Dirichlet conditions u(r, 0) = 0 and u(r, α) = 0, and the coefficients Aₙ are determined by expanding the boundary data on θ = θ₀ in a sine series.

Connection to Complex Variables: Now we compare this expression to the solution obtained with the aid of the complex variable method. The conformal map \( \displaystyle \quad w = z^{\pi/\alpha} \quad \) transforms the wedge to the upper half-plane. In the upper half-plane, harmonic functions can be represented as real parts of analytic functions. The general form of an analytic function in the upper half-plane (or wedge) is: \[ f(z) = \sum_{n=1}^\infty a_n z^{n\pi/\alpha} \] \[ u(r, \theta) = \sum_{n=1}^\infty a_n r^{n\pi/\alpha} \sin\left( \frac{n\pi \theta}{\alpha} \right) \] This matches the separated solution exactly — the powers r^{n\pi/\alpha} and angular modes \sin(n\pi\theta/\alpha) or \cos(n\pi\theta/\alpha) arise naturally in both approaches.

The general solution is: \[ u(r, \theta) = \sum_{n=1}^\infty A_n r^{n\pi/\alpha} \sin\left( \frac{n\pi \theta}{\alpha} \right) \] This function satisfies the Dirichlet conditions u(r, 0) = 0 and u(r, α) = 0, and the coefficients Aₙ are determined by expanding the boundary data on θ = &theta:₀ in a sine series.

Connection to Complex Variables Now compare this to the complex variable method: The conformal map \( \displaystyle \quad w = z^{\pi/\alpha} \quad \) transforms the wedge to the upper half-plane. In the upper half-plane, harmonic functions can be represented as real parts of analytic functions. The general form of an analytic function in the upper half-plane (or wedge) is: \[ f(z) = \sum_{n=1}^\infty a_n z^{n\pi/\alpha} , \] whose real part is:

This function satisfies the Dirichlet conditions u(r, +0) = 0 and u(r, α −0) = 0, and the coefficients A_n are determined by expanding the boundary data on θ = θ_0 in a sine series. 🧠 Connection to Complex Variables Now compare this to the complex variable method: The conformal map w = z^{\pi/\alpha} transforms the wedge to the upper half-plane. In the upper half-plane, harmonic functions can be represented as real parts of analytic functions. The general form of an analytic function in the upper half-plane (or wedge) is: \[ f(z) = \sum_{n=1}^\infty a_n z^{n\pi/\alpha} \] whose real part is:

Philosophical Insight: The separation of variables builds the solution from orthogonal modes — it's spectral. The complex variable method builds the solution from analytic structure — it's geometric. But both converge to the same harmonic function, just viewed through different lenses.    ■

• \( \displaystyle \quad \hat{e}_r \quad \) is the unit vector in the radial direction;
• \( \displaystyle \quad \hat{e}_\theta \quad \) is the unit vector in the angular direction (perpendicular to \( \displaystyle \quad \hat{e}_r , \quad \) pointing counterclockwise).

page 157 of Carrier's book    ■

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