The Poisson Integral Formula on Disc #
Main results #
Theorems poisson_integral_of_harmonicOn_disc_continuousOn_closedDisc and
poisson_integral_of_harmonicOn_disc_continuousOn_closedDisc_ker_re:
Every function u : ℂ → ℝ harmonic on the disc with radius R and center 0, and
continuous on the closed disc, can be represented as
u(z) = 1/(2π) ∫_0^{2π} (R² - |z|²) / |R * exp (it) - z|² * u(R * exp (it)) dt
= 1/(2π) ∫_0^{2π} Re((R * exp (it) + z) / (R * exp (it) - z)) * u(R * exp (it)) dt,
for z in the disc.
Theorem poisson_integral_of_diffContOnCl_disc and poisson_integral_of_diffContOnCl_disc_ker_re:
Every function f : ℂ → E ℂ-differentiable on the disc with radius R and center 0, and
continuous on the closed disc, with values in a complex Banach space E, can be represented as
f(z) = 1/(2π) ∫_0^{2π} (R² - |z|²) / |R * exp (it) - z|² • f(R * exp (it)) dt
= 1/(2π) ∫_0^{2π} Re((R * exp (it) + z) / (R * exp (it) - z)) • f(R * exp (it)) dt,
for z in the disc.
Implementation Notes #
The proof follows from
- Cauchy Integral Formula,
- Cauchy-Goursat Theorem,
- a harmonic function is the real part of a holomorphic function on a disc,
- Lebesgue's Dominated Convergence Theorem.
References #
[Rudin, Real and Complex Analysis (Theorem 11.9)][rudin2006real]
Tags #
ℂ-differentiable function, harmonic function, Poisson integral.
R * exp (t * I) is not equal to any z in the disc of radius R, centered at 0.
R ^ 2 - star z * w ≠ 0, for z in the disc with radius R and center 0,
and for w in the closed disc.
If f is ℂ-differentiable on a disc centered at zero,
then ζ ↦ f (r * ζ) is differentiable at z for r in (0,1) and z in the closed disc.
Cauchy's integral formula applied to f ℂ-differentiable on a disc centered at 0,
at the point r*z, for r in (0,1) and z in the disc.
Cauchy's integral formula for ℂ-differentiable functions on a disc centred at 0,
evaluated at scaled points r * z with r ∈ (0,1).
If f is ℂ-differentiable on a disc centered at zero, then
ζ ↦ (star z / (I * (R ^ 2 - star z * ζ))) • f (r * ζ) is differentiable at w
in the closed disc with radius R and center 0, for r in (0,1).
We apply the Cauchy-Goursat theorem to the function
ζ ↦ (star z / (I * (R ^ 2 - star z * ζ))) • (f (r * ζ)))
on the circle of radius R, centered at 0.
An auxiliary identity that will be used in the integrand of the Cauchy-Goursat theorem.
The Cauchy-Goursat theorem for a disc centered at 0 implies the integral of a
ℂ-differentiable function against a conjugate Cauchy kernel vanishes.
We put together cauchy_integral_formula_scaled_disc and
vanishing_goursat_integral_scaled_disc.
For a ℂ-differentiable function f : ℂ → E on a disc centered at 0, f(r*z) equals the
integral of f(r*R*e^{it}) against the Poisson kernel, where r ∈ (0,1) and z is in the disc.
For a harmonic function u on a disc with radius R, centered at 0,
u(r*z) equals the integral of u(r*R*e^{it}) against the Poisson kernel,
where r ∈ (0,1) and z is in the disc.
We bound t ↦ ‖k (R * exp (t * I)) • f (r * R * exp (t * I))‖, for
k continuous on the circle of radius R and center 0,
and f continuous on the closed disc of radius R and center 0.
For a sequence rₙ → 1 with rₙ ∈ (0,1), the integral of
t ↦ k(R*e^{it}) • f(rₙ*R*e^{it}) on [0 , 2π] converges to the integral of
t ↦ k(R*e^{it}) • f(R*e^{it}) on [0 , 2π], when f is continuous on the
closed disc of radius R and center 0, and k is continuous on the circle of radius R
and center 0, by Lebesgue's Dominated Convergence Theorem.
The Poisson kernel is continuous on the circle.
If rₙ tends to 1, then f (rₙ * z) tends to f z, for z in a disc centered at 0,
when f is continuous on the closed disc.
Poisson integral formula for harmonic functions on a disc:
A function u harmonic on a disc with radius R and center 0,
and continuous on the closed disc, satisfies
u(z) = (1/2π) ∫_0^{2π} (R² - |z|²) / |R*e^{it} - z|² u(R*e^{it}) dt for z in the disc.
Poisson integral formula for ℂ-differentiable functions on a disc:
A function f : ℂ → E ℂ-differentiable on a disc with radius R and center 0,
and continuous on the closed disc, satisfies
f(z) = (1/2π) ∫_0^{2π} (R² - |z|²) / |R*e^{it} - z|² f(R*e^{it}) dt for z in the disc.
Poisson integral formula for harmonic functions on a disc:
A function u : ℂ → ℝ harmonic on a disc with radius R and center 0, and
continuous on the closed disc, satisfies
u(z) = (1/2π) ∫_0^{2π} Re((R*e^{it} + z) / (R*e^{it} - z)) * u(R*e^{it}) dt
for z in the disc.
Poisson integral formula for ℂ-differentiable functions on a disc:
A function f : ℂ → E ℂ-differentiable on a disc with radius R and center 0,
and continuous on the closed disc, satisfies
f(z) = (1/2π) ∫_0^{2π} Re((R*e^{it} + z) / (R*e^{it} - z)) • f(R*e^{it}) dt
for z in the disc.
Poisson integral formula for harmonic functions on a disc:
A function u : ℂ → ℝ harmonic on a disc with radius R and center 0,
and continuous on the closed disc, satisfies
u(z) = (1/2π) ∫_0^{2π} Re((R*e^{it} + z) / (R*e^{it} - z)) * u(R*e^{it}) dt
for z in the disc.
Poisson integral formula for ℂ-differentiable functions on a disc:
A function f : ℂ → E ℂ-differentiable on a disc with radius R and center 0,
and continuous on the closed disc, satisfies
f(z) = (1/2π) ∫_0^{2π} Re((R*e^{it} + z) / (R*e^{it} - z)) • f(R*e^{it}) dt
for z in the disc.