Anharmonic oscillators revisited

Mathematics of Computation

Caboz³

1989

Self Cite

Abstract.We consider the incomplete hyperelliptic integralwith a > 0, A2 > 0, n > 2, where X belongs to the connected component of {x|A2X2 + Anzn < a} containing the origin. Continuing previous work on the complete hyperelliptic integral, we express in this paper H (a, X) as a convergent series of hypergeometric type. A brief survey of some applications to algebraic equations and mechanics is then given.

Section: Figure Lmentioning

confidence: 99%

“…(2) represents the time t to go from the origin to the current point on the X-axis [5]. For Pn given by Eq.…”

Section: í211mentioning

confidence: 99%

Incomplete Hyperelliptic Integrals and Hypergeometric Series

Loiseau¹,

Mathematics of Computation

Caboz³

1989

Self Cite

“…Equation (25) of the present paper gives much simpler results. The values of a and ß in "hypergeometric form" are derived from formulae [12] obtained by the Lagrange-Bürmann Theorem; see also [5].) (Identification of (35), (36), (37), respectively, with (32), (33), (34) is carried out in [12], using transformation relations for Gaussian hypergeometric functions established by Goursat [9].…”

Section: Appendixmentioning

confidence: 99%

Hyperelliptic integrals and multiple hypergeometric series

Loiseau¹,

Caboz³

1988

Math. Comp.

Abstract.We consider the complete hyperelliptic integral Using a recent result concerning the Taylor expansion of the 6-Dirac function, we write J(a) as a power series of a parameter involving a and the Ajt'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 Pn is less than or equal to one. The region of convergence is then studied and a few particular cases are detailed.

“…The incomplete integral H(a, X) of Eq. (2) represents the time t to go from the origin to the current point on the X-axis [5]. For Pn given by Eq.…”

mentioning

confidence: 99%

“…For Pn given by Eq. (5), the time t is expressed by the series (18) and may be written in the form oo (22) t=xylpxp, p=0…”

mentioning

confidence: 99%

Incomplete hyperelliptic integrals and hypergeometric series

Loiseau¹,

Caboz³

1989

Math. Comp.

Self Cite

We consider the incomplete hyperelliptic integral \[ H ( a , X ) = ∫ 0 X d x a − λ 2 x 2 − λ n x n H(a,X) = \int _0^X {\frac {{dx}}{{\sqrt {a - {\lambda _2}{x^2} - {\lambda _n}{x^n}} }}} \] with a > 0 a > 0 , λ 2 > 0 {\lambda _2} > 0 , n > 2 n > 2 , where X belongs to the connected component of { x | λ 2 x 2 + λ n x n > a } \{ x|{\lambda _2}{x^2} + {\lambda _n}{x^n} > a\} containing the origin. Continuing previous work on the complete hyperelliptic integral, we express in this paper H ( a , X ) H(a,X) as a convergent series of hypergeometric type. A brief survey of some applications to algebraic equations and mechanics is then given.