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Die Determinantenmethode zur Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung

On the determinantal method for calculating the characteristic exponent of the finite Hill differential equation

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Summary

The characteristic exponent ν of the finite Hill differential equation

$$y''(x) + \left( {\lambda + \sum\limits_{\kappa = 1}^k {(2t_\kappa ) \cos (2\kappa x)} } \right) y(x) = 0$$

can be evaluated from the relations

$$\sin ^2 \left( {\frac{\pi }{2}v} \right) = \frac{{\pi ^2 }}{4} \det C^{(0)} \det S^{(0)}$$

or

$$\cos ^2 \left( {\frac{\pi }{2}v} \right) = \det C^{(1)} \det S^{(1)} ,$$

whereS (μ) andC (μ) are certain infinite band matrices. According to Mennicken [3] the convergence of the infinite determinants can be accelerated by splitting up suitable infinite products. In the present paper this method is discussed under numerical aspects, moreover the formulas for the infinite products are simplified in such way that the complex Gamma-function is no longer needed. Finally, the presented determinental method is compared with other methods by means of some numerical examples.

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Literatur

  1. Hill, G.W.: On the part of the motion of the lunar perigee, which is a function of the mean motions of the sun and the moon. Acta Math.8, 1–36 (1886)

    Google Scholar 

  2. Magnus, W.: Infinite determinants associated with Hill's equation. Pacific J. Math.5, Suppl. 2, 941–951 (1955),

    Google Scholar 

  3. Mennicken, R.: On the convergence of infinite Hill-type determinants. Arch. Rational Mech. Anal.30, 12–37 (1968)

    Google Scholar 

  4. Mennicken, R., Wagenführer, E.: Über die Konvergenz verallgemeinerter Hillscher Determinanten. Math. Nachr.72, 21–49 (1976)

    Google Scholar 

  5. Mennicken R., Wagenführer, E.: Numerische Mathematik 1. Reinbek: Rowohlt-Vieweg 1976

    Google Scholar 

  6. Schäfke, F.W., Schmidt, D.: Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung, III. Numer. Math.8, 68–71 (1966)

    Google Scholar 

  7. Wagenführer, E.: Ein Verfahren höherer Konvergenzordnung zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung. Numer. Math.27, 53–65 (1976)

    Google Scholar 

  8. Wagenführer, E., Lang, H.: Berechnung des charakteristischen Exponenten der endlichen Hill-schen Differentialgleichung durch Numerische Integration Numer. Math.32, 31–50 (1979)

    Google Scholar 

  9. Whittaker, E.T., Watson, G.N.: A course of modern analysis. 4th ed., Cambridge: Cambridge University Press 1965

    Google Scholar 

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Wagenführer, E. Die Determinantenmethode zur Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung. Numer. Math. 35, 405–420 (1980). https://doi.org/10.1007/BF01399008

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