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


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


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