expansion of zeta using stieltjes-constants

[behackl]

With the implementation of Stieltjes-constants from #19834 the expansion of the zeta-function can be improved to something like

sage: zeta(s).series(s==1)
1/(s-1) + euler_gamma - stieltjes(1)/2 * (s-1) + stieltjes(2)/6 * (s-1)^2  + ...

(see https://en.wikipedia.org/wiki/Stieltjes_constants).

Depends on #19834

CC: @rwst

Component: symbolics

Author: Benjamin Hackl

Branch/Commit: f1d9bfa

Reviewer: Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/19836

[behackl]

comment:1

I suppose that---given the stieltjes-function is implemented in pynac---this only requires a change in the zeta1_series function in pynac.

I think I can adapt this and make a pull request on github; at least if there is a stieltjes-function.

[rwst]

Dependencies: #19834

[rwst]

comment:2

Yes. Dependency is set. This ticket would then just add doctesting of the expansion.

[behackl]

comment:3

I've created a pull request here. Assuming this gets merged into pynac, the doctest here:
https://github.com/sagemath/sage-prod/blob/master/src/sage/functions/transcendental.py#L81-L82
will fail and has to be removed in the ticket where pynac is updated.

This ticket here would then add the respective doctest of the form

sage: zeta(x).series(x==1, 4)
1*(x - 1)^(-1) + (euler_gamma) + (-1/2*stieltjes(1))*(x - 1) + (1/6*stieltjes(2))*(x - 1)^2 + (-1/24*stieltjes(3))*(x - 1)^3 + Order((x - 1)^4)

Or do you suggest a different procedure?

[rwst]

comment:4

Replying to @behackl:

Or do you suggest a different procedure?

It's fine. Maybe add one of the examples I just asked about in the pull requests.

[rwst]

comment:5

With the changes I get this, correct?

sage -t src/sage/functions/transcendental.py
**********************************************************************
File "src/sage/functions/transcendental.py", line 81, in sage.functions.transcendental.Function_zeta.__init__
Failed example:
    zeta(x).series(x==1, 1)
Expected:
    1*(x - 1)^(-1) + (euler_gamma + log(2) + log(pi) + 2*zetaderiv(1, 0)) + Order(x - 1)
Got:
    1*(x - 1)^(-1) + (euler_gamma) + Order(x - 1)
**********************************************************************

[behackl]

comment:6

Replying to @rwst:

With the changes I get this, correct?

sage -t src/sage/functions/transcendental.py
**********************************************************************
File "src/sage/functions/transcendental.py", line 81, in sage.functions.transcendental.Function_zeta.__init__
Failed example:
    zeta(x).series(x==1, 1)
Expected:
    1*(x - 1)^(-1) + (euler_gamma + log(2) + log(pi) + 2*zetaderiv(1, 0)) + Order(x - 1)
Got:
    1*(x - 1)^(-1) + (euler_gamma) + Order(x - 1)
**********************************************************************

Exactly, that is the doctest I mentioned above.

[behackl]

Author: Benjamin Hackl

[behackl]

Commit: f1d9bfa

[behackl]

Branch: u/behackl/functions/zeta-stieltjes-doc

[behackl]

comment:7

I've adapted the documentation of zeta(s) in order to reflect the special expansion at s==1; everything else has already happened on other tickets. :-)

---

New commits:

bdac71f	19834: implement symbolic Stieltjes constants
0e1162b	Merge tag '7.1.beta1' into symbolics/stieltjes-constants
f1d9bfa	mention stieltjes-constants in documentation of zeta