Advertisement

Il Nuovo Cimento B (1971-1996)

, Volume 31, Issue 2, pp 257–271 | Cite as

The optimum-index method

  • W. M. Frank
Article
  • 14 Downloads

Summary

A convenient method is presented for the evaluation in the asymptotic limit of the ratio of two entire functions of the same order and type. The coefficients in the power series must be positive, which restricts our applications to scattering theory to repulsive potentials only.

Метод оптимальных индексов

Резюме

Общепринятый метод применяется для вычисления в асимптотическом пределе отношения двух целых функций одинакового порядка и типа. Коэффициенты в степенных рядах должны быть п]qoложительными, что ограничивает наше применение к теории рассеяния тольк дляо потенциалов отталкивания.

Riassunto

Si presenta un metodo adatto per stimare nel limite asintotico il rapporto fra due funzioni intere dello stesso ordine e tipo. Nelle serie di potenze si possono avere coefficienti positivi, col che si restringono le applicazioni relative alla teoria della diffusione solo a potenziali repulsivi.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    The quantities “order» and «type» of an entire function are defined in eqs. (9) and (10) of the text. Roughly speaking the entire functionf(z) grows for asymptotically large |z| like exp [τ|z|ϱ], where ϱ and τ are respectively the order and type off(z).Google Scholar
  2. (2).
    W. M. Frank andD. J. Land:Journ. Math. Phys.,11, 2041, 2058 (1970).ADSCrossRefGoogle Scholar
  3. (3).
    G. Polya andG. Szego:Aufgaben und Lehrsatze aus der Analyse (New York, N. Y., 1945).Google Scholar
  4. (4).
    Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964), p. 504.Google Scholar
  5. (5).
    Op. cit. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964), p. 257.Google Scholar
  6. (6).
    Op. cit. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964), p. 504.Google Scholar
  7. (7).
    Op. cit. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964), p 375.Google Scholar
  8. (8).
    Op. cit. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964), p. 377.Google Scholar
  9. (9).
    The vanishing of every second coefficient makes the present prescription impossible, but poses no problem as only some isolated coefficients vanish.Google Scholar
  10. (10).
    See,e.g., ref. (2).ADSCrossRefGoogle Scholar
  11. (11).
    See,e.g.,R. G. Newton:Journ. Math. Phys.,1, 319 (1960).ADSCrossRefMATHGoogle Scholar
  12. (12).
    F. Calogero andM. Cassandro:Nuovo Cimento,37, 760 (1965).CrossRefGoogle Scholar
  13. (13).
    See,e.g.,W. M. Frank,D. J. Land andR. M. Spector:Rev. Mod. Phys.,43, 36 (1971).MathSciNetADSCrossRefGoogle Scholar
  14. (14).
    The Jost function is known to be an entire function of the coupling constant and to be analytic ink in a half-plane and in a strip including the real axis if the potential falls exponentially for large distances, and is entire ink if it falls faster than an exponential. See,e.g.,R. G. Newton:op. cit. ADSCrossRefMATHGoogle Scholar
  15. (15).
    The Jost function for the repulsive potential can easily be obtained from the expression given byNewton,op. cit., for the attractive potential.ADSCrossRefMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1976

Authors and Affiliations

  • W. M. Frank
    • 1
  1. 1.Naval Surface Weapan CenterSilver Spring

Personalised recommendations