Example 1:    ■

Example 2: Let us consider the exterio of a circle Ω = {(r, θ) : 𝑎 < r < ∞, 0 ≤ θ < 2π} in the two-dimensional plane ℝ². We are looking for a solution of the Laplace equation in Ω satisfying the Dirichlet boundary condition: where f : ℝ ↦ ℝ is a given smooth periodic function of period 2π.

To solve this boundary value problem, we employ the method of separation of variables. Namely, we seek partial nontrivial solutions of the Laplace equation (written in polar coordinates) \[ \nabla^2 \phi = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} = 0 \] that are represented in product form ϕ(r, θ) = R(r) Θ(θ), where R(r) is a function of radial variable and Θ(θ) is a function of angle variable only. Substituting this product into Laplace's equation, we get \[ \frac{1}{r}\,\frac{\text d}{{\text d}r} \left( r\,\frac{{\text d}R(r)}{{\text d}r} \right) \Theta (\theta ) + \frac{R(r)}{r^2}\, \frac{{\text d}^2 \Theta}{{\text d}\theta^2} = 0 , \] which leads to separation of variables (upon multiplication by r² and division by ϕ(r, θ)): \[ \frac{r}{R(r)}\, \frac{\text d}{{\text d}r} \left( r\,\frac{{\text d}R(r)}{{\text d}r} \right) = - \frac{1}{\Theta (\theta )}\,\frac{{\text d}^2 \Theta}{{\text d}\theta^2} = \lambda . \] So we get two ordinary differential equations with unknown yet parameter λ, \[ r^2 R'' + r\, R' - \lambda\,R = 0 , \qquad \Theta'' + \lambda\,\Theta = 0 . \] Since Θ(θ) must be periodic function of period 2π, we obtain the Sturm--Liouville (S.-L. for short) problem: \[ \begin{cases} \Theta'' + \lambda\,\Theta = 0 , \\ \Theta (\theta ) = \Theta (\theta + 2\pi ) . \end{cases} \] To solve this Sturm--Liouville problem, we need to consider three cases whether λ is negative, equals to zero, or positive. If λ is negative, the general solution of differential equation Θ′′ + λΘ = 0 is \[ \Theta (\theta ) = c_1 e^{\theta\sqrt{-\lambda}} + c_2 e^{-\theta\sqrt{-\lambda}} , \] with some real constants c₁ and c₂. This exponential function cannot be periodic, so we reject this case and claim that the given S,-L, problem has no eigenvalues for negative λ,

When λ = 0, the differential equation Θ′′ = 0 has the general solution \[ \Theta (\theta ) = c_1 + c_2 \theta . \] Since the linear term cannot be periodic, we have to set c₂ = 0; so λ = 0 is an eigenvalue to which corresponds the eigenfunction Θ(θ) = constant.

When λ is a positive number, the general solution of the differential equation Θ′′ + λΘ = 0 is \[ \Theta (\theta ) = c_1 \cos \left( \theta\sqrt{\lambda} \right) + c_2 \sin \left( \theta\sqrt{\lambda} \right) , \] with some real constants c₁ and c₂. This trigonometric function can be periodic with period 2π if and only if λ = n² is a square of a positive integer (why positive, see the previous example). So the solution of this S.-L. problem gives eigenvalues and eigenfunctions: \[ \lambda_n = n^2 , \quad \Theta_n = a_n \cos n\theta + b_n \sin n\theta ,\qquad n=0,1,2,\ldots . \] Substitution of this λ = n² into R-equation yields the Euler ODE \[ r^2 R'' + r\, R' - n^2\,R = 0 , \qquad n= 0,1,2,\ldots . \] We start with n = 0, which is reduced to \[ r^2 R'' + r\, R' = 0 \qquad \iff \qquad r\,\frac{\text d}{{\text d}r} \left( r\,\frac{{\text d}R}{{\text d}r} \right) = 0 . \] This differential equation has two linearly independent solutions R = constant and R = lnr. We reject the latter because logarithm function is unbounded at infinity (the regularity condition requires all functions to be bounded). So we get one function R₀(r) = constant corresponding to eigenvalue λ = 0.

When n > 0, the R-equation has two linearly independent solutions R = rⁿ and R = r⁻ⁿ. We reject the former because rⁿ is unbounded at infinity. This allows us to define the solution of the given Dirichlet problem for exterior of a circle: \[ \phi (r, \theta ) = \frac{a_0}{2} + \sum_{n=1}^{\infty} r^{-n} \left( a_n \cos n\theta + b_n \sin n\theta \right) . \tag{2.1} \] The coefficients of this series are determined from the boundary condition: \begin{align*} \phi (a, \theta &= f (\theta ) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a^{-n} \left( a_n \cos n\theta + b_n \sin n\theta \right) . \end{align*} Since this is a Fourier series for given function f(θ), its coefficients are determined according to Euler--Fourier formulae \[ a_n = \frac{a^n}{\pi} \int_0^{2\pi} f(\theta )\,\cos n\theta \,{\text d}\theta , \qquad b_n = \frac{a^n}{\pi} \int_0^{2\pi} f(\theta )\,\sin n\theta \,{\text d}\theta \tag{2.2} \] We can then substitute these expressions for 𝑎ₙ, bₙ into the series solution (2.1). Comparing this expression with teh similar one of the previous example (interior circle case), we can see that the only difference in the solutions is that r is replaced by r^&minus1, and 𝑎 by 𝑎⁻¹. So we can follow the same procedure, and sum the series explicitly, since in this case the magnitude of the (complex) ratios of the geometric series will be a/r, which is less than one in the exterior of the circle, thus leading to summable series. Then Poisson’s formula in the exterior of the circle will be \[ \phi (r, \theta ) = \left( a^{-2} - r^{-2} \right) \int_0^{2\pi} \frac{f(s)}{a^{-2} - 2 a^{-1} r^{-1} \cos (\theta -s) + r^{-2}}\,{\text d}s . \]

We can also solve this problem using the conformal mapping w = f(z) = 1/z. This function maps points in the z-plane, (x, y), to points in the w-plane, (u, v), by f(x + ⅉy) = (u + ⅉv), where ⅉ (also is denoted by j) is the imaginary unit on the complex plane so ⅉ² = −1. This mapping maps the interior of the unit circle to the exterior of the unit circle in the w-plane.

   ■

Example 3:    ■

Example 4:    ■

Example 5: Let us consider the annulus V = {(r, ϕ) | 𝑎_- ≤ r ≤ 𝑎₊} in the two-dimensional plane ℝ². We are looking for a solution of the Laplace equation in V satisfying the Dirichlet boundary conditions: where f_-, f₊ : ℝ ↦ ℝ are a given smooth periodic functions of period 2π.

To solve this boundary value problem, we employ the method of separation of variables. Namely, we seek partial nontrivial solutions of the Laplace equation (written in polar coordinates) \[ \nabla^2 \phi = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} = 0 \] that are represented in product form ϕ(r, θ) = R(r) Θ(θ), where R(r) is a function of radial variable and Θ(θ) is a function of angle variable only. Substituting this product into Laplace's equation, we get \[ \frac{1}{r}\,\frac{\text d}{{\text d}r} \left( r\,\frac{{\text d}R(r)}{{\text d}r} \right) \Theta (\theta ) + \frac{R(r)}{r^2}\, \frac{{\text d}^2 \Theta}{{\text d}\theta^2} = 0 , \] which leads to separation of variables: \[ \frac{r}{R(r)}\, \frac{\text d}{{\text d}r} \left( r\,\frac{{\text d}R(r)}{{\text d}r} \right) = - \frac{1}{\Theta (\theta )} \,\frac{{\text d}^2 \Theta}{{\text d}\theta^2} = \lambda . \] So we get two ordinary differential equations with unknown yet parameter λ, \[ r^2 R'' + r\, R' - \lambda\,R = 0 , \qquad \Theta'' + \lambda\,\Theta = 0 . \] Since function Θ(θ) must be periodic function of period 2π, we obtain the Sturm--Liouville (S.-L. for short) problem: \[ \begin{cases} \Theta'' + \lambda\,\Theta = 0 , \\ \Theta (0) = \Theta (2\pi ) . \end{cases} \] Solution of this S.-L. problem gives eigenvalues and eigenfunctions: \[ \lambda_n = n^2 , \quad \Theta_n = a_n \cos n\theta + b_n \sin n\theta ,\qquad n=0,1,2,\ldots . \] Substitution of this λ = n² into R-equation yields the Euler ODE \[ r^2 R'' + r\, R' - n^2\,R = 0 , \qquad n= 0,1,2,\ldots . \] We start with n = 0, which is reduced to \[ r^2 R'' + r\, R' = 0 \qquad \iff \qquad r\,\frac{\text d}{{\text d}r} \left( r\,\frac{{\text d}R}{{\text d}r} \right) = 0 . \] This differential equation has two linearly independent solutions R = constant and R = lnr.

When n ≠ 0, the R-equation has two linearly independent solutions R = rⁿ and R = r⁻ⁿ. This allows us to define the solution of the given Dirichlet problem for annulus domain: \[ \phi (r, \theta ) = \frac{1}{2} \left( C_0 + D_0 \ln r \right) + \sum_{n=1}^{\infty} \left( C_n r^n + D_n r^{-n} \right) \cos n\theta + \left( A_n r^n + B_n r^{-n} \right) \sin n\theta . \] The coefficients of this series are determined from the boundary conditions: \begin{align*} \phi (a_{-}, \theta &= f_{-} (\theta ) = \frac{1}{2} \left( C_0 + D_0 \ln a_{-} \right) \\ &\quad + \sum_{n=1}^{\infty} \left( C_n a_{-}^n + D_n a_{-}^{-n} \right) \cos n\theta + \left( A_n a_{-}^n + B_n a_{-}^{-n} \right) \sin n\theta , \\ \phi (a_{+}, \theta &= f_{+} (\theta ) = \frac{1}{2} \left( C_0 + D_0 \ln a_{+} \right) \\ &\quad + \sum_{n=1}^{\infty} \left( C_n a_{+}^n + D_n a_{+}^{-n} \right) \cos n\theta + \left( A_n a_{+}^n + B_n a_{+}^{-n} \right) \sin n\theta , \end{align*} Using Fourier coefficients corresponding to Fourier expansions of given functions f_- and f₊, we obtain \begin{align*} C_0 + D_0 \ln a_{-} &= \frac{1}{\pi} \int_0^{2\pi} f_{-} (\theta )\,{\text d} \theta , \\ C_n a_{-}^n + D_n a_{-}^{-n} &= \frac{1}{\pi} \int_0^{2\pi} f_{-} (\theta )\,\cos (n\theta )\,{\text d} \theta , \quad n=1,2,\ldots , \\ A_n a_{-}^n + B_n a_{-}^{-n} &= \frac{1}{\pi} \int_0^{2\pi} f_{-} (\theta )\,\sin (n\theta )\,{\text d} \theta , \quad n=1,2,\ldots , \\ C_0 + D_0 \ln a_{+} &= \frac{1}{\pi} \int_0^{2\pi} f_{+} (\theta )\,{\text d} \theta , \\ C_n a_{+}^n + D_n a_{+}^{-n} &= \frac{1}{\pi} \int_0^{2\pi} f_{+} (\theta )\,\cos (n\theta )\,{\text d} \theta , \quad n=1,2,\ldots , \\ A_n a_{+}^n + B_n a_{+}^{-n} &= \frac{1}{\pi} \int_0^{2\pi} f_{+} (\theta )\,\sin (n\theta )\,{\text d} \theta , \quad n=1,2,\ldots . \end{align*} Using Mathematica, we solve these systems of linear equations. \begin{align*} C_0 &= \frac{1}{\ln a_{-} + \ln a_{+}}\,\frac{1}{\pi} \left[ \ln a_{-}\, \int_0^{2\pi} f_{+} (\theta )\,{\text d} \theta - \ln a_{+}\, \int_0^{2\pi} f_{-} (\theta )\,{\text d} \theta \right] , \\ D_0 &= \frac{1}{\ln a_{-} + \ln a_{+}} \left[ -\frac{1}{\pi} \int_0^{2\pi} f_{+} (\theta )\,{\text d} \theta + \frac{1}{\pi} \int_0^{2\pi} f_{-} (\theta )\,{\text d} \theta \right] , \end{align*}    ■

1. Burington, R. S. (1940). On the use of Conformal Mapping in Shaping Wing Profiles. The American Mathematical Monthly, 47(6), 362–373. https://doi.org/10.1080/00029890.1940.11990989
2. Carrier, G.F., Krook, M., and Pearson, C.E., Functions of a Complex Variable: Theory and Technique, Society for Industrial and Applied Mathematics, 2005.