9.2 Poisson integral formula

The Poisson integral formula is a powerful tool for solving harmonic function problems on the unit disk. It lets us find harmonic functions inside the disk using their values on the boundary. This connects to earlier topics on harmonic functions and their properties.

The formula uses the Poisson kernel to weight boundary values and integrate them. This gives us a way to solve Dirichlet problems, finding harmonic functions with specific boundary conditions. It's a key technique for working with harmonic functions in complex analysis.

Poisson Integral Formula for Unit Disk

Derivation of Poisson Integral Formula

Top images from around the web for Derivation of Poisson Integral Formula

• 

  understanding a theorem in complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  integration - solving integral with complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  real analysis - Integration by parts in solving Poisson equation - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  understanding a theorem in complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  integration - solving integral with complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Derivation of Poisson Integral Formula

• 

  understanding a theorem in complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  integration - solving integral with complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  real analysis - Integration by parts in solving Poisson equation - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  understanding a theorem in complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  integration - solving integral with complex analysis - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

• Represents a harmonic function on the unit disk using its boundary values
• Assumes u(z) is a harmonic function on the unit disk D and continuous on the closure of D
• Expresses u(z) as an integral of its boundary values weighted by the Poisson kernel P(r, θ)
  • r is the distance from the origin to z
  • θ is the angle between the positive real axis and the line segment from z to a boundary point
• Poisson kernel formula: P(r, θ) = $(1 - r^2) / (1 - 2r \cos(\theta) + r^2)$
• Derivation steps:
  • Express u(z) as the real part of a holomorphic function f(z)
  • Use Cauchy's integral formula to represent f(z) using its boundary values
  • Obtain the imaginary part of f(z), the harmonic conjugate of u(z), using the Cauchy-Riemann equations
  • Relate boundary values of f(z) to boundary values of u(z): $f(e^{i\theta}) = u(e^{i\theta}) + iv(e^{i\theta})$, where v is the harmonic conjugate of u

Poisson Integral Formula and Its Properties

• Final form of Poisson integral formula: $u(re^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} P(r, \theta - t) u(e^{it}) dt$
  • $u(e^{it})$ represents the boundary values of u
• Properties of Poisson kernel:
  • Non-negative and integrates to 1 over the boundary, making it a probability density function
  • As a point approaches the boundary, the Poisson kernel converges to a Dirac delta function centered at the boundary point
• Mean value property: The value of a harmonic function at any point is the average of its values on any circle centered at that point
• Behavior under harmonic conjugation: The Poisson integral of the boundary values of a harmonic function yields its harmonic conjugate (up to a constant)

Applying Poisson Integral to Boundary Problems

Solving Dirichlet Boundary Value Problems

• Dirichlet boundary value problem: Find a harmonic function on a domain that takes prescribed values on the boundary
• To solve a Dirichlet problem on the unit disk using Poisson integral:
  • Obtain the boundary values of the desired harmonic function
  • Compute the Poisson integral of the boundary values: $u(z) = \frac{1}{2\pi} \int_0^{2\pi} P(|z|, \arg(z) - t) f(e^{it}) dt$, where $f(e^{it})$ represents the given boundary values
• The resulting function u(z) is guaranteed to be harmonic on the unit disk and takes the prescribed boundary values
• Example: Find a harmonic function on the unit disk that takes the boundary values $f(e^{i\theta}) = \cos(\theta)$

Practical Considerations and Techniques

• Evaluating the Poisson integral may involve:
  • Numerical integration techniques (trapezoidal rule, Simpson's rule)
  • Series expansions of the boundary values (Fourier series)
• Accuracy and efficiency of the solution depend on the complexity of the boundary values and the chosen integration method
• Example: Solve the Dirichlet problem on the unit disk with boundary values $f(e^{i\theta}) = e^{\cos(\theta)}$ using a Fourier series expansion

Generalizing Poisson Integral to Other Domains

Poisson Integral for Upper Half-Plane

• Represents a harmonic function on the upper half-plane using its boundary values on the real line
• Poisson kernel for the upper half-plane: $P(x, y) = y / (x^2 + y^2)$, where (x, y) is a point in the upper half-plane
• Poisson integral formula for the upper half-plane: $u(x, y) = \frac{1}{\pi} \int_{-\infty}^{\infty} P(x - t, y) f(t) dt$, where f(t) represents the boundary values on the real line
• Example: Find a harmonic function on the upper half-plane that takes the boundary values $f(x) = e^{-x^2}$

Poisson Integral for Ball in Higher Dimensions

• Represents a harmonic function on a ball using its boundary values on the sphere
• Poisson kernel for a ball in n dimensions: $P(r, \theta) = (1 - r^2) / |r - \theta|^n$, where r is a point inside the ball and θ is a point on the boundary sphere
• Poisson integral formula for a ball: $u(r) = \frac{1}{\omega_n} \int_S P(r, \theta) f(\theta) dS(\theta)$
  • $\omega_n$ is the surface area of the unit sphere in n dimensions
  • S is the boundary sphere
  • $f(\theta)$ represents the boundary values
• Example: Find a harmonic function on the unit ball in 3D that takes the boundary values $f(\theta) = \cos(\theta_1)\sin(\theta_2)$

Poisson Integral and Harmonic Functions

Connection between Harmonic Functions and Boundary Values

• Every harmonic function on a domain can be represented by the Poisson integral of its boundary values
• Conversely, the Poisson integral of any continuous function on the boundary yields a harmonic function on the interior
• Poisson kernel is a harmonic function in the interior of the domain
• As a point approaches the boundary, the Poisson kernel converges to a Dirac delta function centered at the boundary point, allowing the Poisson integral to recover the boundary values
• Example: Show that the Poisson integral of the boundary values $f(\theta) = \cos(n\theta)$ on the unit disk yields the harmonic function $u(r, \theta) = r^n \cos(n\theta)$

Relationship to Other Integral Formulas and Green's Function

• Poisson integral formula is a special case of the more general Schwarz integral formula, which represents a holomorphic function on a domain using its real part on the boundary
• Poisson integral is closely related to the Green's function for the Laplace equation
  • Green's function is a fundamental solution that can be used to solve inhomogeneous boundary value problems
  • The Poisson kernel can be obtained by taking the normal derivative of the Green's function on the boundary
• Example: Derive the Green's function for the Laplace equation on the unit disk and show its relationship to the Poisson kernel