EllipticTheta and "rules"

• To: mathgroup at yoda.physics.unc.edu
• Subject: EllipticTheta and "rules"
• From: Richard Muirhead <muirhead at u.washington.edu>
• Date: Mon, 20 Sep 1993 11:33:26 -0700 (PDT)

```   I'm working with a small group studying aspects of "quantum chaos",
and our meanderings have brought us to the use of the elliptic theta
functions.  These functions are, fortunately, defined within
Mma, but none of the crucial properties (various recursion relationships)
seem to be known to Mma!!

For example, define:

theta3[z_,s_]:=EllipticTheta[3,z,E^(I*Pi*s)]
theta4[z_,s_]:=EllipticTheta[4,z,E^(I*Pi*s)]

I would like ... theta3[z,s]-theta4[z+Pi/2,s]... to test True.

But, alas, Mma seems not to know these various relationships, and
must be trained with, I presume, "rules".  And this is where we're
getting into trouble.  In particular, those more complicated
relationships involving connections between the various derivatives
of the elliptic functions have us stumped.

How do we construct the appropriate rules so that the EllipticTheta is
not just a more-or-less useless symbol??

We've read Maeder's relevant chapters, and have some luck with
rules that don't involving relating the derivatives (... and I do
mean just **some** luck...), but we're unable to generalize.

I come from the "procedural programming" paradigm and just haven't
seen the light yet on this "functional" approach (...is that the
right vocabulary??...).  The closer I look at Mma and the more
complicated the problem, the more confused I become.  For me, Mma
has that in common with quantum mechanics and members of the
opposite sex.  And my frustration is not unique based on the comments
from  fellow Mma novices on this quantum chaos project.

So I guess yet another question I have, more vague and I suspect
more difficult to field, is "how does one get the 'functional' message."
Is there a "right" book? (...a right prayer, a right genuflection, etc).

I thank the members of the group for any insights/directions they
might offer on our elliptic function problem.  I have been amazed at the
insightful responses I have received from previous posts.  And I again
extend my sincere thanks.  Maybe if I keep hammering away at this
complex beast, I too will "see the pattern" and then be able to
return the many favors I have received from this community.

Thanks

Rick Muirhead
Physics UW
muirhead at phys.washington.edu

```

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