# 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^{} $u$. ^{1}^{1}Since $u+iv$ is an analytic function^{} of $z=x+iy$,it is clear from the Cauchy-Riemann equations^{} that the function $v(x,y)$ is determined by
$v(x,y)={\displaystyle {\int}_{{z}_{0}}^{z}}{\displaystyle \frac{\partial v}{\partial x}}\mathit{d}x+{\displaystyle \frac{\partial v}{\partial y}}dy={\displaystyle {\int}_{{z}_{0}}^{z}}-{\displaystyle \frac{\partial u}{\partial y}}dx+{\displaystyle \frac{\partial u}{\partial x}}dy,$
where the integral^{} is evaluated over an arbitrary path joining some point ${z}_{0}$ with an arbitrary point $z$ belonging to the unitary open disc $D$. We are concerned to a simply connected domain, so that the function $v(x,y)$ will be single-valued.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

$\frac{1}{2\pi i}}{\displaystyle {\int}_{\gamma}}{\displaystyle \frac{w(\zeta )}{\zeta -z}}\mathit{d}\zeta +{\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int}_{\gamma}}{\displaystyle \frac{\overline{w}(\overline{\zeta})}{\zeta -z}}\mathit{d}\zeta ={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int}_{\gamma}}{\displaystyle \frac{f(\theta )}{\zeta -z}}\mathit{d}\zeta ,$ | (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 ^{2}^{2}From Taylor’s formula
$w(z)=w(0)+{w}^{\prime}(0)z+{\displaystyle \frac{1}{2!}}{w}^{\prime \prime}(0){z}^{2}+O({z}^{3}).$
But on $\gamma $, $\overline{z}=1/\zeta $, so
$\overline{w}(\overline{\zeta})=\overline{w}(0)+{\overline{w}}^{\prime}(0){\displaystyle \frac{1}{\zeta}}+{\displaystyle \frac{1}{2!}}{\overline{w}}^{\prime \prime}(0){\displaystyle \frac{1}{{\zeta}^{2}}}+O\left({\displaystyle \frac{1}{{\zeta}^{3}}}\right)$
and term-by-term integration gives the desired result recalling that
$\frac{1}{2\pi i}}{\displaystyle {\int}_{\gamma}}{\displaystyle \frac{d\zeta}{{\zeta}^{n}(\zeta -z)}}=\{\begin{array}{cc}1,\hfill & ifn=0,\hfill \\ 0,\hfill & otherwise.\hfill \end{array$
the second one is equal to $\overline{w}(0)$. Let $\overline{w}(0)=a-ib$, thus (3) becomes

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

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

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

whence

$2a={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int}_{\gamma}}{\displaystyle \frac{f(\theta )}{\zeta}}\mathit{d}\zeta ={\displaystyle \frac{1}{\pi i}}{\displaystyle {\int}_{0}^{2\pi}}f(\theta )\mathit{d}\theta .$ | (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),

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

the aimed Schwarz formula.^{3}^{3}It is possible to prove that, if $f(\theta )$ satisfies Hölder condition, then the function $w(z)$ given by (6) will be continuous in $\overline{D}$. Such a condition is less restrictive than the requirement of the existence of a bounded derivative^{}.

## 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

$\mathrm{\Re}w(z)\equiv u(\rho ,\varphi )={\displaystyle \frac{1}{2\pi}}{\displaystyle {\int}_{0}^{2\pi}}{\displaystyle \frac{(1-{\rho}^{2})f(\theta )}{1-2\rho \mathrm{cos}(\theta -\varphi )+{\rho}^{2}}}\mathit{d}\theta .$ | (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.^{4}^{4}See [1]. 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.^{5}^{5}For a discussion of Neumann problem, see [2].

## References

- 1 O. D. Kellog, Foundations of Potential Theory, Dover, 1954.
- 2 G. C. Evans, The Logarithmic Potential, Chap. IV, New York, 1927.

Title | Schwarz and Poisson formulas |
---|---|

Canonical name | SchwarzAndPoissonFormulas |

Date of creation | 2013-03-22 16:05:58 |

Last modified on | 2013-03-22 16:05:58 |

Owner | perucho (2192) |

Last modified by | perucho (2192) |

Numerical id | 12 |

Author | perucho (2192) |

Entry type | Theorem |

Classification | msc 30D10 |

Defines | Schwarz formula |

Defines | Poisson formula |