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Les fonctions hypercylindriques dans l’espace à n+2 dimensions. (French) JFM 47.0348.02

Setzt man im vierdimensionalen Raume

x=ϱsinθsinψ,y=ϱsinθcosψ,z=ϱcosθ,t=t,

und sucht harmonische Funktionen von der Form

U=e μt cosνψV(ϱ,θ)

(μ,ν Konstant), so kommt man auf die vom Verf. betrachteten Hyper-Zylinderfunktionen V· Sie genügen der Differentialgleichung

ϱ 2 2 V ϱ 2 +(1-ω 2 ) 2 V ω 2 +2ϱV ϱ-2ωV ω+μ 2 ϱ 2 V-ν 2 1-ω 2 V=0,

wenn cosθ=ω gesetzt wird. Sie stehen in einfacher Beziehung zu den Appelschen hypergeometrischen Funktionen zweier Veränderlichen.

Die zweite Note überträgt diese Betrachtungen auf höhere Räume. (IV 13.)