Asymptotische Entwicklung Beispiel Essay

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Between hard Difference essay football and soccer

In der Mathematik und ihren Anwendungen bezeichnet asymptotische Analyse eine Methode um das Grenzverhalten von Funktionen zu klassifizieren, indem man nur den wesentlichen Trend des Grenzverhaltens beschreibt.

Beschreibung des asymptotischen Verhaltens[Bearbeiten | Quelltext bearbeiten]

Das asymptotische Verhalten von Funktionen lässt sich mit einer Äquivalenzrelation beschreiben. Sind beispielsweise und reellwertige Funktionennatürlicher Zahlenn, so lässt sich eine Äquivalenzrelation definieren durch

genau dann wenn

.

Die Äquivalenzklasse von besteht aus allen Funktionen , bei denen der relative Fehler zu beim Grenzübergang gegen 0 strebt. Diese Definition lässt sich unmittelbar auf Funktionen einer reellen oder komplexen Veränderlichen übertragen sowie auf den Fall gegen , wobei die Annäherung an oft nur über eine Teilmenge erfolgt, z. B. im Reellen von links oder von rechts, bzw. im Komplexen in einem Winkelbereich, oder über eine vorgegebene diskrete Menge.

Einige Beispiele für asymptotische Resultate[Bearbeiten | Quelltext bearbeiten]

Landau-Notation[Bearbeiten | Quelltext bearbeiten]

Eine nützliche Notation zur Beschreibung der Wachstumsklassen ist die Landau-Notation, die ursprünglich von Paul Bachmann stammt, aber durch Edmund Landau bekannt gemacht wurde. Eine wichtige Anwendung der Landau-Notation ist die Komplexitätstheorie, in der asymptotische Laufzeit und Speicherverbrauch eines Algorithmus untersucht werden.

Die einfachste Art, diese Symbole zu definieren, ist die folgende: und sind Klassen von Funktionen mit den Eigenschaften

Für alle
Für alle

wird in der Regel aus dem Kontext klar. Weiter schreibt man häufig statt das Folgende: .

Asymptotische Entwicklung[Bearbeiten | Quelltext bearbeiten]

Unter einer asymptotischen Entwicklung einer Funktion versteht man die Darstellung der Funktion als nicht notwendigerweise konvergente Reihe. Die Partialsummen einer solchen Reihe brauchen nicht zu konvergieren, liefern aber in der Nähe von gute Näherungen für den Funktionswert. Das bekannteste Beispiel einer asymptotischen Entwicklung ist die stirlingsche Reihe als asymptotische Entwicklung für die Fakultät. Definieren lässt sich eine solche Entwicklung mit Hilfe einer asymptotischen Folge als

mit .

Falls die asymptotische Entwicklung nicht konvergiert, gibt es für jedes Funktionsargument einen Index , bei dem der Approximationsfehler

am kleinsten wird; Hinzufügen weiterer Terme verschlechtert die Approximation. Der Index der besten Approximation wird bei asymptotischen Entwicklungen aber umso größer, je näher bei liegt.

Asymptotische Entwicklungen treten insbesondere bei der Approximation gewisser Integrale auf, beispielsweise mittels der Sattelpunktmethode. Das asymptotische Verhalten von Reihen lässt sich darauf oft mit Hilfe der eulerschen Summenformel zurückführen.

Literatur[Bearbeiten | Quelltext bearbeiten]

  • Erdélyi, A.: Asymptotic Expansions, New York: Dover, 1987.
  • Berg, L.: Asymptotische Darstellungen und Entwicklungen, Berlin: VEB Deutscher Verlag der Wissenschaften, 1968.

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