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Eine neue Erweiterung der Laplace-Transformation. I, II. (German) JFM 67.0396.01
Das Formelpaar (Laplace-Transformation und Umkehrung) $$ f(s)=\int\limits_0^\infty e^{-st}F(t)dt,\quad \dfrac{F(t+0)+F(t-0)}2=\lim\limits_{\lambda\to\infty} \dfrac1{2\pi i} \int\limits_{\beta-\lambda i}^{\beta+\lambda i} e^{ts}f(s)ds \tag1 $$ erweitert Verf. folgendermaßen: $$ f(s) = \int\limits_0^\infty e^{-\frac12 st} W_{k+\frac12,m}(st)(st)^{-k-\frac12} F(t) dt, \tag2 $$ $$ F(t)=\lim\limits_{\lambda\to\infty} \dfrac{\varGamma(1-k+m)}{2\pi i\varGamma(1+2m)} \int\limits_{\beta-\lambda i}^{\beta+\lambda i} e^{\frac12 ts} M_{k-\frac12,m}(ts)(ts)^{k-\frac12} f(s)ds, \tag3 $$ wobei $W_{k,m}(z)$ und $M_{k,m}(z)$ die Whittakerschen Funktionen sind. Wegen $$ W_{-m+\frac12,m}(z) = z^{\tfrac12-m}e^{-\tfrac{z}2},\quad M_{-m-\frac12,m}(z) = z^{\tfrac12+m}e^{\tfrac{z}2} $$ 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.
Reviewer: Doetsch, G.; Prof. (Freiburg im Breisgau)