DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Poisson kernel on the unit disc

For $z \in D = {∣ z ∣ < 1}$ and $t \in R$ , the Poisson kernel of the unit disc is

$P (z, e^{i t}) : = \frac{1 - ∣ z ∣^{2}}{∣ e^{i t} - z ∣^{2}} .$

If $z = r e^{i ϕ}$ with $0 \leq r < 1$ , then

$P (z, e^{i t}) = \frac{1 - r^{2}}{1 - 2 r \cos (t - ϕ) + r^{2}} = : P_{r} (t - ϕ) .$

The second formula is just the first one written in polar coordinates. It is the form used in the one-variable estimates and in Harnack's inequality.

Used by

Nothing. This result depends on no other item in the library.