Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Hyperelliptic integrals and multiple hypergeometric series


Authors: Jean-Francis Loiseau, Jean-Pierre Codaccioni and Régis Caboz
Journal: Math. Comp. 50 (1988), 501-512
MSC: Primary 33A25; Secondary 33A35
DOI: https://doi.org/10.1090/S0025-5718-1988-0929548-0
MathSciNet review: 929548
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the complete hyperelliptic integral

$\displaystyle J(a) = \int_{\alpha (a)}^{\beta (a)} {\frac{{dx}}{{\sqrt {a - {P_n}(x)} }},}$

where $ a > 0$ and

$\displaystyle {P_n}(x) = \sum\limits_{k = 2}^n {{\lambda _k}{x^k},}$

with $ {\lambda _2} > 0,[\alpha ,\beta ]$ being the connected component of $ \{ x\vert{P_n}(x) \leq a\} $ containing the origin.

Using a recent result concerning the Taylor expansion of the $ \delta $-Dirac function, we write $ J(a)$ as a power series of a parameter involving a and the $ {\lambda _k}$'s.

We prove this series to be a sum of multiple hypergeometric series which reduces to a single term when the number of odd monomial terms in $ {P_n}$ is less than or equal to one.

The region of convergence is then studied and a few particular cases are detailed.


References [Enhancements On Off] (What's this?)

  • [1] R Caboz, J. P. Codaccioni & F. Constantinescu, "Taylor series for the Dirac function on perturbed surfaces with applications to mechanics," Math. Methods Appl. Sci., v. 7, 1985, pp. 416-425. MR 827201 (87m:70028)
  • [2] R. Caboz & J. F. Loiseau, "Lien entre période et hauteur normalisée à l'intérieur du puits de potentiel pour l'oscillateur anharmonique," C.R. Acad. Sci. Paris, v. 296, 1983, pp. 1753-1756.
  • [3] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. MR 0590943 (58:28707)
  • [4] J. P. Codaccioni & R. Caboz, "Anharmonic oscillators and generalized hypergeometric functions," J. Math. Phys., v. 25, 1984, pp. 2436-2438. MR 751528 (85f:70024)
  • [5] J. P. Codaccioni & R. Caboz, "Anharmonic oscillators revisited," Internat. J. Non-Linear Mech., v. 20, 1985, pp. 291-295. MR 804639 (87a:70014)
  • [6] A. Erdélyi, Higher Transcendental Functions, Vol. 1 (Bateman Manuscript Project), Mc-Graw-Hill, New York, 1953.
  • [7] H. Exton, Multiple Hypergeometric Functions and Applications, Ellis Horwood, Chichester, 1976. MR 0422713 (54:10699)
  • [8] I. M. Gelfand & G. E. Shilov, Generalized Functions, Academic Press, New York, 1964. MR 0166596 (29:3869)
  • [9] E. Goursat, "Sur l'équation différentielle linéaire qui admet pour intégrale la série hypergéometrique," Ann. Ecole Norm. Sup., v. 10, suppl. Paris, 1881.
  • [10] J. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980.
  • [11] M. Lakshmanan & J. Prabhakaran, "The energy level of an $ {x^6}$ anharmonic oscillator," Lett. Nuovo Cimento, v. 7, 1973, pp. 689-692.
  • [12] J. F. Loiseau, Contribution à l'Étude des Solutions Exactes et Approchées de l'Oscillateur Anharmonique Monômial en Mécanique Classique (Thèse d'Etat), Université de Pau et des Pays de l'Adour, 1986.
  • [13] M. S. Spiegel, Formules et Tables de Mathématiques (Série Schaum), McGraw-Hill, New York, 1983.
  • [14] H. M. Srivastava & P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood, Chichester, 1985. MR 834385 (87f:33015)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 33A25, 33A35

Retrieve articles in all journals with MSC: 33A25, 33A35


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1988-0929548-0
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society