[om] A&S as authority.

Richard Fateman fateman at cs.berkeley.edu
Thu Nov 6 00:21:50 CET 2003

Some cautions.

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.


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.
>>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.
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