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Re: [Axiom-developer] Patches
From: |
Waldek Hebisch |
Subject: |
Re: [Axiom-developer] Patches |
Date: |
Mon, 4 Jun 2007 17:31:46 +0200 (CEST) |
Martin Rubey wrote:
> Waldek Hebisch <address@hidden> writes:
>
> > > But the behaviour is consistent for Gamma, Bessel and Polygamma. It is
> > > not
> > > difficult to change this behaviour to leaving the derivative unevaluated,
> > > but I'm not sure whether that would really be better. If you are
> > > absolutely sure, please let me know as soon as possible.
> >
> > Yes, currently we produce mathematically incorrect result. In principle
> > user
> > may get wrong results even if input does not contain explicit derivative.
>
> Oh? How is that?
>
Consider something like naive test for holonomic functions: compute
derivatives up to some fixed order and check for linear dependence.
Such test would immediatly conclude that besselK(a, x) is holonomic
as a function of a. I did not check this but I think that
besselK(a, x) is not holonomic as a function of a, and certainly
we would get wrong differential equation. Once you have differential
equation in hand you can do a lot of transformations.
Of course, currently Axiom has no support for holonomic functions.
But in few places we use derivatives: changing variables in
integrals, computing Laplace transforms. It is quite possible that
Axiom never uses derivative of bessel function with respect to
parameter. But checking this would be a substantial ongoing effort.
> > Once we get better support for special functions this may be very serious
> > problem.
>
> Probably. By the way: most (probably all) special functions would be covered
> by
> my favourite would-be category/domain hierarchy of differentially algebraic
> functions. Then we could say something like
>
> polygamma(a, x)$HOLO(???)
>
> and get the corresponding differential equation.
>
Hmm, gamma and consequently also polygamma(a, x) as a function of x
is differential transcendental. Also handling of non-holonomic
differentially algebraic functions seem to be a research problem
-- do you have some interesting results here?
> > > How about polygamma? should D(polygamma(x, x), x) throw an error? I
> > > guess so.
> > > But if we follow you, Bessel* should leave the derivative with respect to
> > > the
> > > first argument - i.e., leave it unevaluated.
>
> > polygamma(a, x) has sensible definition also for non-integral a, so just
> > leaving D(polygamma(x, x), x) unevaluated is reasonable.
>
> I could not find such a definition. Could you please send me such a
> definition
> or a reference?
>
>From http://mathworld.wolfram.com/PolygammaFunction.html:
A special function which is given by the (n+1) st derivative of the
logarithm of the gamma function Gamma(z)
....
....
psi_n(z) is implemented in Mathematica as PolyGamma[n, z] for positive
integer n . In fact, PolyGamma[nu, z] is supported for all complex nu
(Grossman 1976; Espinosa and Moll 2004).
I do not know which definition the references use, but a derivatives
may be defined for fractional orders via convolution:
{d \over dx}^n f = f*mu_{-n-1}
where
mu_l(x) = x^l/\Gamma(l+1) for x > 0
and mu_l(x) = 0 for x < 0.
This definition of derivative is for non-integral n, for integral
n you get normal derivative as a limit.
The definition above will get function which is analytic in n. Because
analytic functions have strong restictions on possible zeros other
definitions are likely to give the same value.
--
Waldek Hebisch
address@hidden
- [Axiom-developer] Patches, Martin Rubey, 2007/06/02
- Re: [Axiom-developer] Patches, Waldek Hebisch, 2007/06/04
- Re: [Axiom-developer] Patches, Martin Rubey, 2007/06/04
- Re: [Axiom-developer] Patches, Waldek Hebisch, 2007/06/05
- Re: [Axiom-developer] Patches, Martin Rubey, 2007/06/05
- Re: [Axiom-developer] Patches, Waldek Hebisch, 2007/06/05
- Re: [Axiom-developer] Patches, Gabriel Dos Reis, 2007/06/04