did
Posts:
73
Registered:
9/14/05
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Re: Inversion Lerch Phi
Posted:
Apr 15, 2012 6:40 AM
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On 4/15/12 12:01 PM, clicliclic@freenet.de 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.
Did
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