Beispiel: Kelvin-Funktionen

Funktionen > Spezielle Funktionen > Bessel-Funktionen > Beispiel: Kelvin-Funktionen

Zeigen Sie die Beziehung zwischen den Funktionen ber und bei an. Diese Funktionen haben keine skalierten Versionen.

1. Plotten Sie Funktionen ber und bei nullter Ordnung:

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Die beiden Funktionen werden symmetrisch dargestellt.

2. Setzen Sie die Teilstrichwerte zurück, um die x-Achse zu vergrößern, um anzuzeigen, wie sich die beiden Spitzenwerte der Funktionen ber und bei ändern, wenn deren Ordnung erhöht wird:

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3. Verwenden Sie die symbolische Auswertung, um die Beziehungen der Funktionen ber und bei zur Funktion Jn zu zeigen, und erstellen Sie anschließend Diagramme, um die Ergebnisse visuell darzustellen.

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In diesem Beispiel geht es um die Funktion ber, die gleichen Beobachtungen gelten jedoch auch für die Funktion bei.

4. Definieren Sie den Vektor Z1 mithilfe der Funktion ber:

Die Funktionen localmin und localmax erfordern eine Eingabematrix mit zwei Spalten. Übergeben Sie den Vektor Z an die Funktion augment, um eine solche Matrix zu erstellen.

5. Suchen Sie die lokalen Minima der Funktion ber:

Die Funktion ber hat 8 Minima, wenn m=0. Die Funktion ist symmetrisch. Funktionen höherer Ordnung weisen eine andere Anzahl von Minima an anderen Positionen und mit anderen Amplituden auf.

6. Suchen Sie die lokalen Maxima der Funktion ber:

Die Funktion ber hat 7 Maxima, wenn m=0. Die Funktion ist symmetrisch. Funktionen höherer Ordnung weisen eine andere Anzahl von Maxima an anderen Positionen und mit anderen Amplituden auf.

7. Ermitteln Sie die vertikalen Koordinaten der vier rechtsseitigen Minima.

8. Verwenden Sie die Funktion match, um den Index des jeweiligen Minimums zu suchen, und ermitteln Sie dann die entsprechenden horizontalen Koordinaten.

9. Suchen Sie die vertikalen Koordinaten der vier rechtsseitigen Maxima.

10. Verwenden Sie die Funktion match, um den Index des jeweiligen Maximums zu suchen, und ermitteln Sie dann die entsprechenden horizontalen Koordinaten.

Der Bereich der Maxima- und Minima-Amplituden variiert erheblich und erschwert ihre Darstellung in einem Diagramm.

11. Verwenden Sie Markierungen, um das erste Minimum (D0) und das erste Maximum (P0) anzuzeigen:

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Wie erwartet, zeigt der Graph ein zweites Minimum links von der y-Achse.

12. Verwenden Sie Markierungen, um das nächste Minimum (D1) und das nächste Maximum (P1) anzuzeigen:

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Wie erwartet, zeigt der Graph ein zweites Minimum und ein zweites Maximum links von der y-Achse.

13. Verwenden Sie Markierungen, um das nächste Minimum (D2) und das nächste Maximum (P2) anzuzeigen:

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Wie erwartet, zeigt der Graph ein zweites Minimum und ein zweites Maximum links von der y-Achse.

14. Verwenden Sie Markierungen, um das nächste Minimum (D3) und das nächste Maximum (P3) anzuzeigen:

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Wie erwartet, zeigt der Graph ein zweites Maximum links von der y-Achse.

Das Minimum (D3) und seine linksseitige Entsprechung liegen genau auf den Kanten des Bereichs. Daher ist es besser, sie zu ignorieren.

Zugehörige Themen

Bessel-Kelvin-Funktionen

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