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Asymptotische Analysis: Zentrale Sätze Und Anwendungen
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ISBN10: 3662733420
ISBN13: 9783662733424
Publisher: Springer Spektrum
Published: Jun 16 2026
Pages: 158
Language: German
ISBN13: 9783662733424
Publisher: Springer Spektrum
Published: Jun 16 2026
Pages: 158
Language: German
Das Buch basiert auf einer Vorlesung zur asymptotischen Analysis, die der Autor mehrfach an der Universität Stuttgart gehalten hat. Es richtet sich dabei sowohl an Studierende, die asymptotische Methoden kennenlernen wollen, als auch an Anwender, die neue Impulse für ihre Arbeit suchen. Nach einer kurzen Einführung asymptotischer Konzepte werden dabei insbesondere asymptotische Entwicklungen und das Rechnen mit asymptotischen Entwicklungen thematisiert. Standardtechniken zum Beweis asymptotischer Formeln, wie die Methode von Laplace und die Methode des steilsten Abstiegs zur Abschätzung von komplexen Kurvenintegralen, werden an Beispielen diskutiert.
Ein immer wiederkehrendes Motiv ist dabei der Zusammenhang zwischen dem asymptotischen Verhalten einer Folge oder Funktion einerseits und analytischen Fortsetzungen eines über eine Transformation zugeordneten Objektes andererseits. Auf diese Weise werden Erzeugendenfunktionen thematisiert und genutzt, um Rekursionsgleichungen zu lösen. Weiter wird die Theorie der Laplace- und Mellintransformation aufgebaut, um Differentialgleichungen zu verstehen und einfache Abelsche und Taubersche Sätze formulieren zu können. Anschlie end werden Dirichletreihen zur Untersuchung zahlentheoretischer Funktionen und zum Beweis des Primzahlsatzes genutzt. Ergänzend dazu werden in einem letzten Kapitel analytische Störungsresultate für Nullstellen von Polynomen sowie für Eigenwerte behandelt und Differentialgleichungen mit regulären und irregulären singulären Punkten diskutiert und das asymptische Verhalten von Lösungen in der Nähe dieser Punkte untersucht.
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