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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 8, 2013 1:03 AM
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Waldek Hebisch schrieb:
> wrote:

> >
> > According to Gradsteyn & Ryzhik 8.14, am(u,k) possesses an infinite
> > number of branch points which form a two-dimensional lattice in the
> > complex plane:
> >
> > u = 2*m*K + (2*n+1)*K'*#i
> >
> > where K = K(k) and K' = K(k') = K(sqrt(1-k^2)) involve the complete
> > elliptic integral of the first kind. These branch points coincide with
> > poles of the Jacobian elliptic functions sn(u,k) = sin(am(u,k)), cn(u,k)
> > = cos(am(u,k)) and dn(u,k) = sqrt(1 - k^2*sin(am(u,k))^2). However, the
> > amplitude am(u,k) is periodic only along the imaginary axis with period
> > 4*K'; along the real axis it has an additional linear component, in
> > accordance with d/du am(u,k) = dn(u,k), the period here being 2*K.
> >
> > For numerical evaluation, am(u,k) must be restricted to a single-valued
> > function by introducing branch cuts. I looks desirable to me that the
> > pattern of cuts preserves both periods. The straightforward choice is
> > to connect all the branch points in pairs with the cuts approaching the
> > two points at u = +- K'*#i from opposite directions, but this still
> > leaves infinitely many possibilities: cuts aligned with the real axis;
> > cuts aligned with the imaginary axis; and finally diagonal orientations.
> > Once the orientation is decided on, periodicity along both axes would
> > generate all other cuts.
> >
> > Obviously the Mathematica and Maple programmers have placed their branch
> > cuts differently, and the actual placement shouldn't be hard to
> > establish by color-coded 3D plots. I concur with Waldek that continuity
> > of am(u,k) on the real line is desirable, which would preclude cuts
> > crossing the real axis. Continuity should probably be abandoned if it
> > cannot be reconciled with having sn(u,k) = sin(am(u,k)) and cn(u,k) =
> > cos(am(u,k)) hold for all complex u and all complex k.

> Well, this equation can be used to define 'am'. Together with
> value at single point and cuts you get full definition. Note
> that the only problem with solving this equations is at poles
> of 'sn' (which are the same as poles of 'cn'). The other
> potentially troublesome point is 0, but 'sn' and 'cn' never
> vainish together, so this is excluded.
> The nasty part is complex 'k'. Or in notation I prefer 'm'.
> When 'm' varies angle between periods changes so if you
> use halflines in 'z' plane for cuts they will change direction
> with changing 'm'.
> Let me add that in FriCAS I did not impement 'am' precisely
> because cuts are so arbitrary: FriCAS code would have to
> spent effort to get on the defined sheet and then for
> user any fixed choice of cuts may be wrong, so the user
> probably would have to redo the work on cuts to get
> them as he/she needs.
> I wonder what use Did has of 'am' for complex arguments,
> that should give some hints which cuts (if any) would
> be good for him.

Thanks, without means to experiment I preferred to tread carefully.

So there will never be a conflict with sn(u,k) = sin(am(u,k)) and
cn(u,k) = cos(am(u,k)) whatever one's choice of branch cuts: at the cuts
am(u,k) cannot but jump in steps of 2*pi. So Mathematica doesn't just
implement am(u,m) for some unsuitable choice of cuts, it doesn't really
implement the Jacobi amplitude. And so at least for real K, Maple's
am(u,k) can be made continuous on the real line by adding steps of size
2*pi where Re(u) equals 2*m*K, which makes one wonder if Maple's am(u,k)
could in fact be disconnected.

I would consider it a natural choice if the pattern of branch cuts for
complex K and K' moves along with the lattice of branch points. While
this is likely to introduce discontinuities on the real line, continuity
would be preserved along the 'm' diagonal in the complex plane. And I am
not at all pessimistic as to the uses of am(u,k) for a good, fixed
(maybe even standardized) choice of branch cuts (that is, cuts along the
+m, -m, +n, or -n directions, I wouldn't seriously contemplate diagonal
orientations). How else is a CAS going to evaluate INT(dn(u,k),u)?

The Labahn-Humphries paper "Symbolic Integration of Jacobi[an] Elliptic
Functions in Maple" may be of interest here:



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