Schwarz and Poisson formulas


Fundamental boundary-value problems of potential theoryMathworldPlanetmath, i.e. (, 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 formulasMathworldPlanetmathPlanetmath for a circular ( boundary. We shall give Schwarz and Poisson formulas that solve the Dirichlet problemMathworldPlanetmath for a circular domain.

Schwarz formula

Without loss of generality, we shall consider the compactPlanetmathPlanetmath disc D¯:|z|1 in the z-plane, its boundary will be denoted by γ and any point on this one by ζ=eiθ. Let it be required to determine a harmonic functionMathworldPlanetmath u(x,y), which on the boundary γ assumes the values

u|γ=f(θ), (1)

where f(θ) is a continuousMathworldPlanetmathPlanetmath single-valued function of θ. Let v(x,y) be the conjugate harmonic function which is determined to within an arbitrary constant from the knowledge of the functionMathworldPlanetmath u. 11Since u+iv is an analytic functionMathworldPlanetmath of z=x+iy,it is clear from the Cauchy-Riemann equationsMathworldPlanetmath that the function v(x,y) is determined by v(x,y)=z0zvx𝑑x+vydy=z0z-uydx+uxdy, where the integralDlmfPlanetmath is evaluated over an arbitrary path joining some point z0 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


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

w(ζ)+w¯(ζ¯)=2f(θ)onγ. (2)

We define here w¯(ζ)=w(ζ¯)¯ and w¯(ζ¯)=w(ζ)¯. Next, we multiply (2) by 12πidζζ-z and, by integrating over γ, we obtain

12πiγw(ζ)ζ-z𝑑ζ+12πiγw¯(ζ¯)ζ-z𝑑ζ=1πiγf(θ)ζ-z𝑑ζ, (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 22From Taylor’s formula w(z)=w(0)+w(0)z+12!w′′(0)z2+O(z3). But on γ,  z¯=1/ζ, so w¯(ζ¯)=w¯(0)+w¯(0)1ζ+12!w¯′′(0)1ζ2+O(1ζ3) and term-by-term integration gives the desired result recalling that 12πiγdζζn(ζ-z)={1,ifn=0,0,otherwise. the second one is equal to w¯(0). Let w¯(0)=a-ib, thus (3) becomes

w(z)=1πiγf(θ)ζ-z𝑑ζ-a+ib. (4)

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



2a=1πiγf(θ)ζ𝑑ζ=1πi02πf(θ)𝑑θ. (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)=1πiγf(θ)ζ-z𝑑ζ-12πiγf(θ)ζ𝑑ζ+ib=12πiγf(θ)ζ+zζ-zdζζ+ib, (6)

the aimed Schwarz formula.33It is possible to prove that, if f(θ) satisfies Hölder condition, then the function w(z) given by (6) will be continuous in D¯. Such a condition is less restrictive than the requirement of the existence of a bounded derivativePlanetmathPlanetmath.

Poisson formula

If we substitute z=ρeiϕ and ζ=eiθ in (6) and separate the real and imaginary partsDlmfPlanetmath, we find

w(z)u(ρ,ϕ)=12π02π(1-ρ2)f(θ)1-2ρcos(θ-ϕ)+ρ2𝑑θ. (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 assumptionPlanetmathPlanetmath that f(θ) is a piecewise continuous function.44See [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 functionPlanetmathPlanetmath and the idea of conformal mappingMathworldPlanetmathPlanetmath of simply connected domains.55For a discussion of Neumann problem, see [2].


  • 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