By I. M. Yaglom

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Für den zugehörigen Funktionswertf(ZI) findet man nun: f( ZI ) = b0 - bmE m -bo + bm+I E m+l W m+l + . +bnE nw n bm = (l - Em)bo +bm+I Em+lwm+l + ... +bnEnW n Diese Gleichung erlaubt nun eine Abschätzung für den Betrag If(zl) l: If (Z I)1 :::; (1- E m)Ibo1+ E m (lb +1w m 1+ ... +1 m +1 + Ibn w nI) = Ibo l((l - E"'(1-lB)), wobei der Quotient B = (lbm+1 wm+1 1+ ... +Ibnwnl)/Ibol nur von den Koeffizienten bo, b m, ... , bn und der darauf aufbauenden Wahl der Zahl w abhängt. Dabei kann der Wert E> 0 offensichtlich so klein gewählt werden, dass 1 - eB positiv ist.

L. van der Waerden, Algebra I, Berlin 1971, § 64. 22 Casus irreducibilis - die Geburtsstunde der komplexen Zahlen rp = arccos [(~+=os [~RJ. so dass man für die drei Lösungen der reduzierten kubischen Gleichung x 3 + px + q = 0 die Formeln (fürj = 0,1,2) erhält. Mit Algebra hat eine solche Lösung auf der Basis von trigonometrischen Funktionen eigentlich nichts zu tun. Allerdings ist die Fragestellung, nämlich die Lösung einer kubischen Gleichung, zweifellos ein fundamentales algebraisches Problem.

N. Grund ist, dass die Zahl ,,1" auf einen von n Plätzen getauscht werden kann, während für die Zahl ,,2" nur noch n-l Plätze zur Verfügung stehen und so weiter. Insgesamt kombinieren sich diese Möglichkeiten zu der angegebenen Anzahl n·(n- l)· ... ·3·2·1. Die Notation einer Permutation aerfolgt am einfachsten mittels einer Aufzählung ihrer Bilder, das heißt den nach der Vertauschung erreichten Plätzen 0(1), 0(2), ... , o(n). Symbolisch notiert man dafür ( 1 2 a(l) a(2) ... n a(n) ) beziehungsweise-kUrzer (a(1) a(2) ...

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Álgebra Extraordinaria by I. M. Yaglom


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