Handling branch cuts for hypergeometric functions

>>>>> "Stavros" == Stavros Macrakis  writes:

    Stavros> When I get a non-trivial asksign question, I often answer it by
    Stavros> assuming "typical" values for the parameter.  For example, if it asks
    Stavros> the sign of x^3-y^2, I'll consider that in my problem x is roughly
    Stavros> 10^2 and y is roughly 1, so clearly it's positive.

How well would this work in practice?  It seems straightforward for
the simple cases, but then they're simple.  But what about more
complicated situations that actually requires some real thought?

These kinds of things always reminds of an example given in Kahan's
paper Much Ado about Nothing's Sign.  He has an example two fairly
complicated functions and asks if they're equal.  If you plug some
random values, they are equal.  I think that if you do some naive
manipulations, you'll prove that they are equal.  In fact, they are
equal, except for in a small oval region of the plane, which you'd
probably not find by random testing.  The difference arises because of
the branch cuts of the functions involved.

[snip]

    Stavros> Of course, this typical value thing would be much less useful if we
    Stavros> had interval arithmetic, or if assume/is knew how to use interval
    Stavros> analysis, e.g.

I've written an interval arithmetic package that is used by CMUCL's
compiler.  It would be straight-forward to rip that out from CMUCL.
If we're only dealing with rationals, it's straightforward.  If we
have to deal with floats, it becomes much messier.  

So, I could probably be persuaded to do something about interval
arithmetic.  :-) Hooking it into maxima in the way you want is a
different and much more difficult question. :-(

Ray