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Topic: Bug in Jacobian Amplitude?
Replies: 16   Last Post: Apr 13, 2013 1:35 PM

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Posts: 1,245
Registered: 4/26/08
Re: Bug in Jacobian Amplitude?
Posted: Apr 13, 2013 1:35 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply schrieb:
> Waldek Hebisch schrieb:

> >
> > Well, I did not see Jacobi paper so I do not not what his definition
> > was, but DMLF 22.16.1 says
> >
> > am(x,k)=Arcsin(sn(x,k)),
> >
> > where the inverse sine has its principal value when -K<= x <= K and
> > is exended by continuity elsewere. Of course this make sense
> > for real x. If you naively extend this definition to complex
> > plane you violate cn(u,k) = cos(am(u,k)). OTOH cn(u,k) = cos(am(u,k))
> > is 22.16.13, so clearly DMLF editors have relaxed view about
> > consistency of their formulas...
> >
> > Anyway, my point is that 'am' is rather poorly defined in the
> > literature.

> As a multivalued function C->C it is (of course) completely defined, but
> CAS implementors seem to have been terribly lazy about defining a
> "standard" branch. [...]
> If DLMF claims that it is working in an "all function symbols C->C refer
> to single predefined branches only" paradigm, then their equations are
> incompatible and therefore partly wrong. I my experience, 19th century
> literature wasn't generally meant to comply with this paradigm and so
> the branches referred to were often undefined or not uniquely defined. I
> wouldn't expect too much from looking up Jacobi about his choice of
> branch cuts or about branch-cut correct functional equations for his
> amplitude function - not much more than from looking up Jonquière or
> Lerch about branch-cut correct equations for the polylogarithm and
> Lerch's transcendent. Even their complex logarithm was usually
> incompatible with the present definition.

Looking at the posts and the DLMF site I notice that I completely missed
the bit about analytic continuation in your description of formula
22.16.1. Accordingly, their definition is not wrong because DLMF meant
the arc sine to be restricted to a standard branch. However, their
definition is limited (without obvious reason) because it cannot be
applied to complex k (or to real k with k^2 > 1) as their argument range
-K <= x <= K becomes meaningless for complex K(k).

In order to apply their definition to complex K, K', it should suffice
to restrict the region on which analytic continuation is based to a
sufficiently small area around the origin x=0. It must be emphasized,
however, that Did's K(k) for k^2 = 3/4 figuring in Mathematica's
violation of cn(x,k) = cos(am(x,k) was real.

Since you (as perhaps does Mathematica) find that the DLMF definition
leads to a conflict with cn(x,k) = cos(am(x,k) for real K(k), I suspect
that your particular "naive" continuation introduces branch cuts (e.g.
cuts starting or stopping at the branch points x = +-K'*#i) across which
am(x,k) doesn't simply jump by 2*pi. If such 'continuations' are
excluded as invalid (compare the doubled branch cut of 1/2*ln(x^2) which
cut the complex logarithm in two) I would expect that nothing is wrong
with the DLMF definition and with the consistency of their formulas ...


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