#18141: special values of transcendental functions
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Reporter: rws | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.6
Component: symbolics | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by rws):
Replying to [comment:2 behackl]:
> I'm rather motivated to implement some of the special values mentioned
above; and I guess the "correct" place would be
`src/functions/transcendental.py`, wouldn't it?
Yes, where the functions are.
> However, I'm not quite sure of how to tackle the implementation of the
[https://en.wikipedia.org/wiki/Stieltjes_constants Stieltjes constants]. I
think that it would be elegant if the constants were also available in
pynac (such that `zeta(s).series(s==1)` could yield the appropriate series
expansion directly).
To recap GiNaC/Pynac by default implements series by differentiation so a
naive way to get a nice expansion would be to implement special values of
the derivatives like in #17678 with the Bessel functions. With `zetaderiv`
this doesn't look like resulting in something usable:
{{{
sage: zeta(x).series(x==1,2)
1*(x - 1)^(-1) + (euler_gamma + log(2) + log(pi) + 2*zetaderiv(1, 0)) +
(1/2*euler_gamma^2 - 1/24*pi^2 + (euler_gamma + log(pi) + 2*zetaderiv(1,
0))*log(2) + 1/2*log(2)^2 + (euler_gamma + 2*zetaderiv(1, 0))*log(pi) +
1/2*log(pi)^2 + 2*euler_gamma*zetaderiv(1, 0) - zetaderiv(2, 0))*(x - 1) +
Order((x - 1)^2)
}}}
I haven't fiddled with Pynac series yet so I'm naive on how to get a
different output for this. I'm also not exactly at home with complex
analysis.
> The problem with this approach is, that I'm not sure of how to implement
a familiy of constants in pynac---or even just in sage.
In Sage there is `symbolic/constants*` with three source files. Earlier I
had started written about the Pynac side in
https://github.com/pynac/pynac/wiki/%7C-constants
To implement a family of symbolic constants (i.e. indexed) would be a
first.
However, at first glance I see no reason why `Stieltjes` could not be
another function taking only arguments from NN.
> What do you think about moving the part with the Stieltjes constants to
a separate ticket?
It looks like this is a sensible approach.
--
Ticket URL: <http://trac.sagemath.org/ticket/18141#comment:3>
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