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Whittaker–Hill equation

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1: 28.32 Mathematical Applications
§28.32(ii) Paraboloidal Coordinates
is separated in this system, each of the separated equations can be reduced to the WhittakerHill equation (28.31.1), in which A , B are separation constants. …
2: Bibliography U
  • K. M. Urwin and F. M. Arscott (1970) Theory of the Whittaker-Hill equation. Proc. Roy. Soc. Edinburgh Sect. A 69, pp. 28–44.
  • 3: 28.31 Equations of Whittaker–Hill and Ince
    §28.31 Equations of WhittakerHill and Ince
    §28.31(i) WhittakerHill Equation
    and constant values of A , B , k , and c , is called the Equation of WhittakerHill. …
    4: 28.34 Methods of Computation
  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the WhittakerHill equation28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • 5: Bibliography
  • F. M. Arscott (1967) The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.
  • 6: Bibliography W
  • M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.
  • M. I. Weinstein and J. B. Keller (1987) Asymptotic behavior of stability regions for Hill’s equation. SIAM J. Appl. Math. 47 (5), pp. 941–958.
  • E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.
  • E. T. Whittaker (1902) On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.
  • E. T. Whittaker (1964) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge.
  • 7: Bibliography S
  • R. A. Silverman (1967) Introductory Complex Analysis. Prentice-Hall, Inc., Englewood Cliffs, N.J..
  • G. F. Simmons (1972) Differential Equations with Applications and Historical Notes. McGraw-Hill Book Co., New York.
  • R. Sips (1965) Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.
  • J. C. Slater (1942) Microwave Transmission. McGraw-Hill Book Co., New York.
  • I. N. Sneddon (1972) The Use of Integral Transforms. McGraw-Hill, New York.
  • 8: Software Index
    9: Bibliography R
  • S. O. Rice (1954) Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.
  • È. Ya. Riekstynš (1991) Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.
  • W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.
  • W. Rudin (1976) Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.