Get EinfГјhrung in die Mathematik: HintergrГјnde der PDF

By Helmut Koch

ISBN-10: 3540203915

ISBN-13: 9783540203919

ISBN-10: 3540430229

ISBN-13: 9783540430223

Diese Einführung besticht durch zwei ungewöhnliche Aspekte: Sie gibt einen Einblick in die Mathematik als Bestandteil unserer Kultur, und sie vermittelt die Hintergründe der Mathematik vom Schulstoff ausgehend bis zum Niveau von Mathematikvorlesungen im ersten Studienjahr. Die Stoffdarstellung geht vom Aufbau der natürlichen Zahlen aus; der Schwerpunkt liegt aber in den exakten Begründungen der Zahlenbegriffe, der Geometrie der Ebene und der Funktionen einer Veränderlichen. Dabei werden alle Sätze bis hin zum Hauptsatz der Algebra vollständig bewiesen. Der klare Aufbau des Buches mit Stichwortregister wichtiger Begriffe erleichtert das systematische Lernen und Nachschlagen. Die zweite Auflage enthält teilweise ausführliche Darstellungen für die Lösungen der zahlreichen Übungsaufgaben. Da viele Aspekte zur Sprache kommen, die so weder im Unterricht noch im Studium behandelt werden, ergänzt die Einführung excellent den Vorlesungsstoff für Lehramtskandidaten und Diplomstudenten.

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A different characterization of fq is presented in Kairies and Muldoon [1982]. 27 Other formulas involving q-Bessel functions are contained in Jackson [1904a-d, 1908], Ismail and Muldoon [1988], Rahman [1987, 1988c, 1989b,c], and Swarttouw and Meijer [1994]. It was pointed out by Ismail in an unpublished preprint in 1999 (rewritten for publication as Ismail [2003c]) that J~3) (x; q) was actually introduced by Jackson [1905a], contrary to the claim in Swarttouw [1992] that a special case of it was first discovered by Hahn [1953] and then in full generality by Exton [1978].

Br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. , if Re( -a) > ml + ... 1) that F [a, r+1 r b+b l ml , ... , br b 1, ... , r + mr , . 1] - 0 - , Re (-a) > ml + ... 3) These formulas are particularly useful for evaluating sums that appear as solutions to some problems in theoretical physics such as the Racah coefficients. They were also used by Gasper [1981b] to prove the orthogonality on (0,27f) of certain functions that arose in Greiner's [1980] work on spherical harmonics on the Heisenberg group.

4) The bilateral q-integral is defined by 1 00 f(t) dqt = (1 - q) f n= -00 [f(qn) + f( _qn)] qn. 5) when restricted. 14) that 1 f is suitably t x-I ((tq; q)oo ) dqt, R ex> 0 , y r--I- 0 ,- 1,- 2, ... , tqY;qoo which clearly approaches the beta function integral B q(x, y ) = 1 o B(x, y) = 11 t x - 1 (1 - t)y-l dt, Re x, Re y > 0, (1117) .. 8) 24 Basic hypergeometric series as q ----+ 1-. 1) in the q-integral form 2¢1 ) _ fq(c) ( a b. c. 10) where Iarg(l- z)1 < 7r and Re c> Re b > O. The q-integral notation is, as we shall see later, quite useful in simplifying and manipulating various formulas involving sums of series.

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EinfГјhrung in die Mathematik: HintergrГјnde der Schulmathematik by Helmut Koch

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