Laplace-Transformation by Walter Ameling (auth.)

By Walter Ameling (auth.)

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2 Der Ableitungssatz fiir die Originalfunktion Der fUr praktische Anwendungen - insbesondere zur Lasung von Differentialgleichungen - wichtigste Satz ist neben dem Satz liber die Linearkombination der Ableitungssatz. Wie aus dem vorangegangenen Abschnitt zu ersehen ist, kommt der Abbildung der Differentiation bei der Lasung von Differentialgleichungen und Differentialgleichungssystemen eine besondere Bedeutung zu. Unter der Voraussetzung, da£ die zu f(t) geharende Bildfunktion fb (s) bekannt ist, soil nun die Laplace-Transformierte der nach t differenzierten Originalfunktion ['(t) = d~it) bestimmt werden.

40 2. Dbergang zur Laplace-Transformation 1m Bildbereich pflegen die mathematischen Beziehungen zwischen den interessierenden Gr6~en bei der Behandlung von Problemen einfacher zu sein als im Originalbereich. Es ist darum zweckma~ig, ein Problem erst in den Bildbereich zu ubertragen, dort zu 16sen und schliemich in den Originalbereich zurUckzutransformieren. Bei dem ersten und letzten Schritt finden haufig die oben erwahnten Tabellen Anwendung. Es ist also gelungen, eine zufriedenstellende Analogie zwischen den Verhaltnissen einer in einem endlichen Interval!

Das Produkt dieser Summe mit dem Abstand der Ordinaten untereinander, also mit ~ , liefert den Wert 2n • f(t). Bei der Grenzwertbetrachtung flir T sich zunachst ¢('A). 3) ¢('A) = f fer) eiA(I-r) dr < 00 -+ 00 andert 2. Obergang zur Laplace-Transformation 38 bleibt. 'A = ~ zu multiplizieren. (I-T) ] dt d'A. Hierin braueht f(t) nun nieht mehr eine periodische Funktion zu sein. 6) f(t) = _1 211" I _= ei wl [ J= fer) e- _= jwT ctrJdW. 7) = J fer) e- jwT dr = fb(w) -= Bildfunktion oder Spektralfunktion von f(t) genannt.

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