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Eine neue Erweiterung der Laplace-Transformation. I, II. (German) JFM 67.0396.01

Das Formelpaar (Laplace-Transformation und Umkehrung)

f(s)= 0 e -st F(t)dt,F(t+0)+F(t-0) 2=lim λ 1 2πi β-λi β+λi e ts f(s)ds(1)

erweitert Verf. folgendermaßen:

f(s)= 0 e -1 2st W k+1 2,m (st)(st) -k-1 2 F(t)dt,(2)
F(t)=lim λ 𝛤(1-k+m) 2πi𝛤(1+2m) β-λi β+λi e 1 2ts M k-1 2,m (ts)(ts) k-1 2 f(s)ds,(3)

wobei W k,m (z) und M k,m (z) die Whittakerschen Funktionen sind. Wegen

W -m+1 2,m (z)=z 1 2-m e -z 2 ,M -m-1 2,m (z)=z 1 2+m e z 2

geht (2), (3) für k=-m in (1) über. – Über die Koexistenz von (2) und (3) werden Sätze bewiesen, die teils von Voraussetzungen über F(t), teils von solchen über f(s) ausgehen und bekannten Sätzen über die Laplace-Transformation analog sind. -Vgl. auch die frühere Arbeit des Verf. in Proc. Akad. Wet. Amsterdam 43 (1940); 599-608, 702-711 (F. d. M. 66, 523 (JFM 66.0523.*)). Doetsch.