[om] A&S as authority.

[Bruce Miller]

Richard Fateman wrote:
> Some cautions.

Indeed, it is tricky business.

As I understand it, there is no "The" branch cuts, merely 
conventional ones, occasionally even obviously-preferable ones,
and/or sets of cuts that maximize the number of identities that
are valid without having to note exceptions.  As Prof. Davenport,
and others, have noted, A&S isn't completely consistent here.

I'm not sure I have the final word on how far the DLMF editors
want to go in this direction; The asymptotics folks in control,
while granting that the issue is important, like to wave thier
hands and mumble `analytic continuation' :>
And certainly when you get into the multi-variable/parameter
very-non-elementary functions, it becomes, um, harder.

BTW: The planned completion for DLMF is late '04.

 
> 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.
>>>   
>>
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> 
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> 


-- 
bruce.miller at nist.gov
http://math.nist.gov/~BMiller/

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