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Schwarz and Poisson formulas
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(Theorem)
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Fundamental boundary-value problems of potential theory, i.e., the so-called Dirichlet and Neumann problems occur in many branches of applied mathematics such as hydrodynamics, elasticity theory and electrodynamics. While solving the two-dimensional problem for special types of boundaries is likely to present serious computational difficulties, it is possible to write down formulas for a circular boundary. We shall give Schwarz and Poisson formulas that solve the Dirichlet problem for a circular domain.
Without loss of generality, we shall consider the compact disc $\overline{D}:|z|\leq 1$ 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
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(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 $u$ . 1Then the function
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
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(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
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(3) |
which, by Harnack's theorem, is equivalent 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 the second one is equal to $\overline{w}(0)$ . Let $\overline{w}(0)=a-ib$ , thus (3) becomes
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(4) |
By setting $z=0$ in (4), we get
whence
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(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),
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(6) |
the aimed Schwarz formula.3
If we substitute $z=\rho\: e^{i\phi}$ and $\zeta=e^{i\theta}$ in (6) and separate the real and imaginary parts, we find
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(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 represents the solution under the assumption that $f(\theta)$ is a piecewise continuous function.4 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
- 1
- O. D. Kellog, Foundations of Potential Theory, Dover, 1954.
- 2
- G. C. Evans, The Logarithmic Potential, Chap. IV, New York, 1927.
Footnotes
- 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
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.
- 2
- From Taylor's formula
But on $\gamma$ , $\overline{z}=1/\zeta$ , so
and term-by-term integration gives the desired result recalling that
- 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.
- 4
- See [1].
- 5
- For a discussion of Neumann problem, see [2].
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| Also defines: |
Schwarz formula, Poisson formula |
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Cross-references: conformal mapping, mapping function, region, piecewise, solution, imaginary parts, derivative, bounded, real, Cauchy's integral formula, Harnack's theorem, boundary condition, class, simply connected, open, unitary, path, integral, Cauchy-Riemann equations, clear, analytic function, conjugate harmonic function, function, single-valued, continuous, harmonic function, point, disc, compact, without loss of generality, domain, circular, Dirichlet problem, formulas, boundaries, occur in, Neumann problems, potential theory
There are 3 references to this entry.
This is version 9 of Schwarz and Poisson formulas, born on 2006-07-21, modified 2008-12-08.
Object id is 8163, canonical name is SchwarzAndPoissonFormulas.
Accessed 4885 times total.
Classification:
| AMS MSC: | 30D10 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Representations of entire functions by series and integrals) |
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Pending Errata and Addenda
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