Exact results for Kerr black holes: shadow areas and perimeters, capture of slow particles, and escape from near extremal horizons

We give exact results for several observables of Kerr black holes that have so far been computed numerically, by fits, or only at isolated parameter values. (i) The area of the shadow seen by an equatorial observer, for every spin, in complete elliptic integrals, together with its small-spin series and its near-extremal expansion A = 16π + 15√3 + (3π/√2)√(1−a) + …; the shadow area of an extremal black hole at every inclination; and a single short integral for arbitrary spin and inclination. (ii) The perimeter of the shadow, equal to 18√3 M for a maximally spinning hole seen edge-on. (iii) The capture cross-section of a Kerr black hole for slow particles arriving perpendicular to the spin, σ = [7π + π√(1−a²) + 16√(1+a) E(2a/(1+a))] M²/v², the corresponding angular-momentum accretion in closed form, and the isotropically averaged cross-section and spin-down at maximal spin in elementary closed form. (iv) Escape probabilities for photons emitted isotropically just outside the horizon of an extremal Kerr–Newman black hole at any latitude, P = 1/2 − arcsin(k)/(2π) − k/4, and from the innermost stable circular orbit at any charge, together with escaping energy fractions and escape probabilities for massive particles. Known special cases are recovered and credited. All results were checked against independent high-precision numerical computations. AI disclosure: This work was carried out with substantial assistance from an AI system, Claude (Anthropic), used through the Claude Code software, under the author's direction. The AI system proposed research directions, wrote and ran the computer-algebra and numerical code, performed the derivations and checks, searched the literature, and drafted the text. The author directed the work and is responsible for its publication. Novelty assessments are based on literature searches and may be incomplete.