Er:Re: Er:Re: [Bug-gsl] discontinuity in dilog function

[Jim McElwaine <J dot N dot McElwaine at damtp dot cam dot ac dot uk>]

Hi Gerard
I went back to 
http://savannah.gnu.org/cgi-bin/viewcvs/gsl/gsl/specfunc/dilog.c
But now it's saying the most recent changes to dilog.c were 
Fri Oct 3 09:27:58 2003
Where is the lastest version?
Jim


Gerard Jungman writes:
 > On Tue, 2004-09-14 at 12:23, Jim McElwaine wrote:
 > > 
 > > It is now appears to be fine for |q|<pi. 
 > > I've tested it for continuity and satisfying  d/dz = log(1-z)/z
 > > 
 > > It still discontinuous for |q|>pi. I imagine this is easy to fix.
 > 
 > Hmmm. But I added tests for that case too, and it passes.
 > I just added a couple more to be absolutely sure; they're
 > in CVS. Can you check again and tell me what you are seeing?
 > 
 > 
 > > I related problem is the definition that is used
 > > 
 > > \int log(1-z)/z dz
 > > 
 > > This has a branch point at z=1 but expressing z = re^{i\theta} is no
 > > use for resolving this so a branch cut is necesary
 > 
 > Agreed, Li_2(z) needs a branch cut, which the
 > implementation puts in the usual place, [1,+infinity).
 > 
 > > If it used the Abramowitz and Stegun definition of
 > > \int log(z)/(1-z) dz then the integral would be well defined for all
 > > \theta using z = re^{i\theta}
 > 
 > Just to make sure we are not talking past each other,
 > what I have implemented is Li_2(z), which is not the
 > Spence function.
 >   Li_2(x) = \sum_1^\infty x^k / k^2
 >   Spence(x) = \sum_1^\infty (1-x)^k / k^2
 > 
 > But Spence cannot avoid the branch cut. The branch
 > point must be at x = 0, the reflection/translation
 > of the branch point for Li_2. The usual place to
 > put the branch cut for Spence is (-infinity,0],
 > which is a reasonable place to put it and also
 > happens to be the reflection/translation of the
 > usual Li_2 branch cut. So it is fair to simply
 > say
 >   Spence(z) = Li_2(1-z)
 > for all complex z, i.e. the analytic properties
 > of each side match exactly.
 > 
 > It is true that the Riemann surface for Spence is
 > topologically the same as that for ln(z), which makes
 > that structure more familiar. But I don't think anything
 > magical happens when going from Li_2 to Spence.
 > Am I missing something?
 > 
 > -- 
 > Gerard Jungman <jungman@lanl.gov>
 > Los Alamos National Laboratory
 > 

-- 
Dr. Jim McElwaine   
H0.15, Department of Applied Maths and Theoretical Physics
Centre for Mathematical Sciences, Cambridge University  
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