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The Poisson integral on the unit disc

Definition

Let $φ : \partial D \to R$ be continuous. Its Poisson integral is the function $P [φ] : D \to R$ defined by

$P [φ] (z) : = \frac{1}{2 π} \int_{0}^{2 π} P (z, e^{i t}) φ (e^{i t}) d t,$

where $P (z, e^{i t})$ is the Poisson kernel of The Poisson kernel on the unit disc.

If $z = r e^{i ϕ}$ , the same formula reads

$P [φ] (r e^{i ϕ}) = \frac{1}{2 π} \int_{0}^{2 π} P_{r} (ϕ - t) φ (e^{i t}) d t .$

Remarks

The boundary datum is written on the unit circle itself, not as a $2 π$ -periodic real function. The angle variable in the integral is only a parametrization.

Depends on

The Poisson kernel on the unit disc

Used by

Dependency tree · one level

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Sources

Jeremy Orloff, MIT 18.04 Topic 5: Introduction to Harmonic Functions (standard reference, not scraped)
Sigurdur Helgason, MIT 18.112 Lecture 16: Harmonic Functions (standard reference, not scraped)