ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 07 May 2020 15:26:51 +0200How to compute the alternating Hurwitz zeta function?https://ask.sagemath.org/question/51298/how-to-compute-the-alternating-hurwitz-zeta-function/Hi all!
What is the best way to compute the alternating Hurwitz zeta function with Sage?
Sage has an implementation of the Hurwitz zeta function, hurwitz_zeta(s,x), where s and x are complex, but not for
the alternating Hurwitz zeta function.
There is the formula https://dlmf.nist.gov/25.11#E35 albeit with significant restrictions on the domain of s and x.
Which is a reliable way to implement hurwitz_alt_zeta(s,x) for general complex s and x, based on Sage functions?
Wed, 06 May 2020 15:23:44 +0200https://ask.sagemath.org/question/51298/how-to-compute-the-alternating-hurwitz-zeta-function/Comment by Sébastien for <p>Hi all!</p>
<p>What is the best way to compute the alternating Hurwitz zeta function with Sage?</p>
<p>Sage has an implementation of the Hurwitz zeta function, hurwitz_zeta(s,x), where s and x are complex, but not for
the alternating Hurwitz zeta function.</p>
<p>There is the formula <a href="https://dlmf.nist.gov/25.11#E35">https://dlmf.nist.gov/25.11#E35</a> albeit with significant restrictions on the domain of s and x.</p>
<p>Which is a reliable way to implement hurwitz_alt_zeta(s,x) for general complex s and x, based on Sage functions?</p>
https://ask.sagemath.org/question/51298/how-to-compute-the-alternating-hurwitz-zeta-function/?comment=51299#post-id-51299A project by Fredrik Johansson [FunGrim: The Mathematical Functions Grimoire](http://fungrim.org/) is to eventually transform automatically such formula into code which respects the restrictions on the domain, etc. I don't know if it is ready yet for what you want to do or if alternating Hurwitz zeta function is in there. See this recent preprint [arXiv:2003.06181](https://arxiv.org/abs/2003.06181)Wed, 06 May 2020 18:14:41 +0200https://ask.sagemath.org/question/51298/how-to-compute-the-alternating-hurwitz-zeta-function/?comment=51299#post-id-51299Answer by fredrik for <p>Hi all!</p>
<p>What is the best way to compute the alternating Hurwitz zeta function with Sage?</p>
<p>Sage has an implementation of the Hurwitz zeta function, hurwitz_zeta(s,x), where s and x are complex, but not for
the alternating Hurwitz zeta function.</p>
<p>There is the formula <a href="https://dlmf.nist.gov/25.11#E35">https://dlmf.nist.gov/25.11#E35</a> albeit with significant restrictions on the domain of s and x.</p>
<p>Which is a reliable way to implement hurwitz_alt_zeta(s,x) for general complex s and x, based on Sage functions?</p>
https://ask.sagemath.org/question/51298/how-to-compute-the-alternating-hurwitz-zeta-function/?answer=51318#post-id-51318The right-hand side in https://dlmf.nist.gov/25.11#E35 looks like a perfectly fine global definition of the alternating Hurwitz zeta function, with the exception of special points. The restrictions Re(a) > 0 and Re(s) > 0 are just there to make the sum and integral well-defined.
The special points are s = 1 (where you need to compute the limit 0.5 * (digamma((a+1)/2) - digamma(a/2))) and the points where either Hurwitz zeta function is not defined with respect to a, namely nonpositive integer a (except when s is a nonpositive integer too).Thu, 07 May 2020 15:26:51 +0200https://ask.sagemath.org/question/51298/how-to-compute-the-alternating-hurwitz-zeta-function/?answer=51318#post-id-51318