#17237: Consistent complex arguments for log integral
-------------------------------------------------+-------------------------
Reporter: kcrisman | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.4
Component: basic arithmetic | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: Reported upstream. Developers | Work issues:
acknowledge bug. | Commit:
Branch: | Stopgaps:
Dependencies: |
-------------------------------------------------+-------------------------
Comment (by kcrisman):
Some thoughts from FJ:
> Here is a possible solution that seems to work off the real line (but
> I haven't checked all corner cases). You can basically write the
> asymptotic expansion of li as a 2F0 plus an ugly branch correction
> term
(http://functions.wolfram.com/GammaBetaErf/LogIntegral/06/02/0003/).
> Removing most of the ugly branch correction term should do the trick.
> The 2F0 diverges, but mpmath can normalize it.
{{{
def li1(z):
return quad(lambda t: 1/ln(t), [2, z]) + li(2)
def li2(z):
return z * hyp2f0(1,1,1/log(z))/log(z) + pi * sign(im(z)) * 1j
}}}
> The outputs seem to agree:
{{{
mp.dps = 30
for n in range(30):
z=30*(rand()-0.5)+30*(rand()-0.5)*1j
print z, chop(li1(z) - li2(z))
}}}
> (You get some large differences, but they appear to be due to the
> numerical integration.)
> I don't think this is the best way to compute the function, but it
> might be *a* way. It needs to be checked to make sure it's actually
> correct, of course.
--
Ticket URL: <http://trac.sagemath.org/ticket/17237#comment:1>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
