Schwarz and Poisson formulas

Introduction

Fundamental boundary-value problems of potential theory, i.e. (http://planetmath.org/Ie), the so-called Dirichlet and Neumann problems occur in many of applied mathematics such as hydrodynamics, elasticity and electrodynamics. While solving the two-dimensional problem for special of boundaries is likely to present serious computational difficulties, it is possible to write down formulas for a circular (http://planetmath.org/Circle) boundary. We shall give Schwarz and Poisson formulas that solve the Dirichlet problem for a circular domain.

Schwarz formula

Without loss of generality, we shall consider the compact disc $\overline{D}:|z|\le 1$ in the $z-$plane, its boundary will be denoted by $\gamma$ and any point on this one by $\zeta =e^{i\theta }$. Let it be required to determine a harmonic function $u(x,y)$, which on the boundary $\gamma$ assumes the values

${u|}_{\gamma }=f(\theta ),$(1)

where $f(\theta )$ is a continuous single-valued function of $\theta$. Let $v(x,y)$ be the conjugate harmonic function which is determined to within an arbitrary constant from the knowledge of the function $u$. Then the function

$w(z)=u(x,y)+iv(x,y)$

is an analytic function for all values of $z\in D$. We shall suppose that $w(z)\in C(\overline{D})$ the class of continuous functions. Therefore, we may write the boundary condition (1) as

$w(\zeta )+\overline{w}(\overline{\zeta })=2f(\theta )\mathit{\hspace{1em}}on\gamma .$(2)

We define here $\overline{w}(\zeta )=\overline{w(\overline{\zeta })}$ and $\overline{w}(\overline{\zeta })=\overline{w(\zeta )}$. Next, we multiply (2) by $\frac{1}{2\pi i}\frac{d\zeta }{\zeta -z}$ and, by integrating over $\gamma$, we obtain

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{w(\zeta )}{\zeta -z}}𝑑\zeta +{\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{\overline{w}(\overline{\zeta })}{\zeta -z}}𝑑\zeta ={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta -z}}𝑑\zeta ,$(3)

which, by Harnack’s theorem, is to (2). Notice that the first integral on the left is equal to $w(z)$ by Cauchy’s integral formula, and for the same reason the second one is equal to $\overline{w}(0)$. Let $\overline{w}(0)=a-ib$, thus (3) becomes

$w(z)={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta -z}}𝑑\zeta -a+ib.$(4)

By setting $z=0$ in (4), we get

$a+ib={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta }}𝑑\zeta -a+ib,$

whence

$2a={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta }}𝑑\zeta ={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int }_0^{2\pi }}f(\theta )𝑑\theta .$(5)

As one would expect, $b$ is left undetermined because the conjugate harmonic function $v(x,y)$ is determined to within an arbitrary real constant. Finally we substitute $a$ from (5) in (4),

$w(z)={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta -z}}𝑑\zeta -{\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta }}𝑑\zeta +ib={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_{\gamma }}f(\theta ){\displaystyle \frac{\zeta +z}{\zeta -z}}{\displaystyle \frac{d\zeta }{\zeta }}+ib,$(6)

the aimed Schwarz formula.

Poisson formula

If we substitute $z=\rho e^{i\varphi }$ and $\zeta =e^{i\theta }$ in (6) and separate the real and imaginary parts, we find

$\mathrm{\Re }w(z)\equiv u(\rho ,\varphi )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{(1-{\rho }^2)f(\theta )}{1-2\rho \cos (\theta -\varphi )+{\rho }^2}}𝑑\theta .$(7)

This is the Poisson formula (so-called also Poisson integral), which gives the solution of Dirichlet problem. It is possible to prove that (7) also the solution under the assumption that $f(\theta )$ is a piecewise continuous function. It is also possible to generalize the formulas obtained above so as to make them apply to any simply connected region. This is done by introducing a mapping function and the idea of conformal mapping of simply connected domains.