[om] A&S as authority.
Bill Naylor
Bill.Naylor at mcs.vuw.ac.nz
Thu Nov 6 00:35:22 CET 2003
The CD transc3 (available from the Extended CD set) aspires to provide the
multivalued versions of the symbols defined in transc1 (as far as A&S is
concerned, those are the ones with capital first letter), that may be
worth a look?
Bill
>
> if you want to (say) look at a recasting of sine/cosine/ etc
> from A&S in the form of recommendations for defining
> relations and branch cuts, you could look at the
> (relatively modern) spec for Common Lisp. I suspect there
> are similar definitions for Java, given the interest that
> Guy Steele Jr had in CL, and then Java.
>
> But the caution is this: those prescriptions are an attempt
> at finding a mapping from a numerical input to an appropriate
> (single-valued) result. I do not know if OM currently
> (or aspires to) provide definitions that incorporate the
> inherent multi-valuedness of (say) log in the complex plane.
> If OM does have such aspirations to multi-valuedness, then
> definitions become far more subtle. Not all functions have
> such simple characterizations of Riemann surfaces and branch
> cuts as log. The LambertW function is a particularly
> nasty example of something rather different.
>
> Even without multiple-valuedness, you get into strange situations
> like what is the meaning of sin(x+y)? X+y is not a number
> at all. By some measures, as long as your definition is a
> kind of "symbolic continuation" of the numerical sine function,
> it might be plausible.
> But there might be many possible such functions.
>
> Cheers
> RJF
>
>
>
>
>
> Bill Naylor wrote:
>
> >On Wed, 5 Nov 2003, Richard Fateman wrote:
> >
> >
> >
> >>Date: Wed, 05 Nov 2003 08:10:47 -0800
> >>From: Richard Fateman <fateman at cs.berkeley.edu>
> >>Reply-To: om at openmath.org
> >>To: om at openmath.org
> >>Subject: Re: [om] Bad bugs in trig CD
> >>
> >>I have not seen enough of the DLMF recently to know how it
> >>is going, (the only full chapter I saw was on the Airy functions)
> >>but how about this:
> >>figure out what exactly it is that you wish to refer to
> >>in A&S as a definition, and copy it into the CD. You could
> >>also say that you believe it to be consistent with the
> >>definition in A&S and DLMF, but you don't require people
> >>to then go find one of those references. There are many
> >>(equivalent) definitions of sine, cosine, etc. Pick one.
> >>e.g. sine (etc) could be infinite series, good for real or
> >>complex, and as these things go, fairly constructive.
> >>
> >>
> >
> >I was looking at A&S chapter 4 last night, and realised there where many
> >different possible definitions of most of these objects. A&S pick one as
> >a definition and then count the others as properties (which is fair
> >enough!). If we were to follow A&S though, it would be nice to have FMPs
> >describing the defining relations. In the case of the inverse functions
> >all of the definitions are given by integrals, which impose conditions on
> >the paths of integrations, e.g.:
> >
> >arcsin(z) = \int^z_0 \frac{dt}{(1-t^2)^{0.5}
> >The path of integration must not cross the real axis
> >
> >There is no way in OpenMath to talk about paths of integration yet, (at
> >least not with the present set of CDs, though I did see one on some jp web
> >site, maybe that should be submitted: Nobuki Takayama, is that your CD?)
> >
> >We could only right these FMPs if this sort of CD was to be accepted by
> >the OpenMath society.
> >
> >Bill
> >
> >
> >
> >>RJF
> >>
> >>PS, I find this vaguely amusing since the need to define
> >>what is meant by such functions was one of the points I raised
> >>at OM meeting 1 or 2, adjacent to some ISSAC meeting.
> >>
> >>
> >--
> >om at openmath.org - general discussion on OpenMath
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> >
> >
>
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