On 4/15/12 12:01 PM, email@example.com wrote: > Sure, one can always switch between formulae whose domains of validity > are nonoverlapping by means of IF statements or constructs like the > Heaviside function. In particular, your formular seems to supply an > additive correction to the Hurwitz zeta function when its second > argument goes "out of range". But this means introducing nonanalytic > objects into complex analysis ...
You are right. If I remember correctly, you can split my formula into two purely analytic ones, with a condition on the range of the argument of z-1. Heaviside's function is not less analytical than a IF statement, it is a IF statement under disguise.
> Actually, for the polylogarithm inversion formula, it is computationally > simpler to just switch between the two versions in the Wikipedia > article. It is still simpler to introduce > > CuteLog(z) = Log(z) if z not in ]0;-1], > = -Log(1/z) if z not in ]-1;-inf[, > > which remains an analytic function (there is no contradiction where the > domains overlap), and to compute > > polylog(s,z) + (-1)^s*polylog(s,1/z) > = (2*I*Pi)^nu/GAMMA(s) * Zeta(0,1-s,1/2+CuteLog(-z)/I/2/Pi) > > for all complex z. CuteLog differs from Log by an infinitesimall small > relocation of the branch cut.
Thanks for the tip.
BTW, Axel's formula is made valid on the branch cut subtracting (there only) 2*I*Pi/z^a (check numerically with both MMA and Maple, and analytically for some special values with MMA).
The question is who is right: Maple hygergeometric analytic equivalent, or Maple numerics and MMA numerics and analytic? 1 againt 3, but democracy shouldn't play any role in Mathematics.
I'm still interested in the general formula for all s.