Jacobian zeta function; Z(\phi|m)?
- To: mathgroup at smc.vnet.net
- Subject: [mg17444] Jacobian zeta function; Z(\phi|m)?
- From: Jason.Wm.Mitchell at uc.edu (Jason W. Mitchell)
- Date: Sun, 9 May 1999 04:43:51 -0400
- Organization: University of Cincinnati, ECE/CS News Server
- Sender: owner-wri-mathgroup at wolfram.com
Hello,
I would like to know what algorithm Mathematica (v3.0) uses to calculate
the Jacobian zeta function; JacobiZeta[\phi,m] or Z(\phi|m), where
\phi = JacobiAmplitude[u,m] or \phi = am(u|m). After searching the
/usr/local/mathematica tree, I could find no file containing a definition
for JacobiZeta[\phi,m] so I presume it is 'compiled' into the kernel...
I have tested Mathematica's built-in algorithm against a zeta function
constructed, in Mathematica, from theta functions, i.e. [get ready for
more LaTeX]
Z(u|m) = \frac{1}{\theta_3{}^2(0)} \frac{\theta_4'(v)}{\theta_4(v)}
with
v = \frac{\pi}{2K(m)} u,
\theta_3{}^2(0) = \frac{2K(m)}{\pi},
using the more modern labeling of theta functions as with Mathematica, as
well as using both forms of the expansion in terms of elliptic integrals
of the first and second kind from Byrd & Friedman, i.e.
Z(\phi|m) = E(\phi|m) - \frac{E(m)}{K(m)} F(\phi|m)
or
Z(u|m) = E(am(u|m)|m) - \frac{E(m)}{K(m)} u.
Neither of the three algorithms match the values of the built-in zeta
function; notice that u = am^{-1}(\phi|m) = F(\phi|m). The built-in
Z(\phi|m) generally out performs, in terms of accuracy, the constructed,
algebraically equivalent functions, but performs significantly better
near the quarter period(s); it also does much better with large arguments.
Thank you,
Jason
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