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

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Waldek Hebisch

Posts: 267
Registered: 12/8/04
Re: Bug in Jacobian Amplitude?
Posted: Apr 7, 2013 3:43 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply 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.

Waldek Hebisch

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