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