General Solution of the Taylor Series Coefficient Problem
in the Theory of Jacobian Elliptic Functions
Karl Wodnar
Institut of astronomy - University of Vienna, Austria
Elliptic Functions are an inevitable ingredient of several theories in
celestial mechanics, see e.g. the interesting work of Williams (1992),
and in dynamics in general. In the course of dealing with a generalized
standard map as proposed by Ichtiaroglou in Wodnar et al. (1996) in
the context of applying non-integrability proofs, we found the present
theory.
In the well known collection of formulas Abramowitz & Stegun (1984) on
page 229 the leading terms of the Taylor expansion of the classical three
Jacobian elliptic functions sn, cn, dn in the default argument (not
the modulus) are given up to seventh order with the following remark added:
`No formulae are known for the general coefficient in these series.'
Also other more recent formula encyclopedias do not go beyond listing
some leading terms of the expansion considered, neglecting the question
of general coefficients.
The present work solves this problem completely also for the nine other
related elliptic functions in a rather concise way. Beside the theoretical
interest of such a result, there is also a new practically applicable
method for numerical calculation of the functions in question at hand,
which - for convergence reasons - is being used on a quarterperiod
and then is transferred to the whole domain by the well known symmetries.