
Re: Bug in Jacobian Amplitude?
Posted:
Apr 13, 2013 1:35 PM


clicliclic@freenet.de 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 branchcut correct functional equations for his > amplitude function  not much more than from looking up Jonquière or > Lerch about branchcut 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 ...
Martin.

