did
Posts:
77
Registered:
9/14/05


Re: Inversion Lerch Phi
Posted:
Apr 13, 2012 5:11 PM


On 4/13/12 10:48 PM, Axel Vogt wrote: > I do not yet agree (for the versions: I used Maple 16): > > MMA just has the the problem to numerical evaluate the example, e. > And I do not want to assume that MMA is right. > > And the final example f2  e is just a reformulation of that for e: > since f2 will be 0 it reduces to that. > > Would you mind to check MMA 7 with z = 2  1e1000 * I?
Without sufficient precision, MMA will "see" this z on the branch cut:
z = 2  I*10^(1000); a = 1/2; Block[{$MaxExtraPrecision = 100}, N[ HurwitzLerchPhi[1/z, 1, a]  HurwitzLerchPhi[z, 1, a] + 1/a  (1/z)^a*Pi/Sin[Pi*(1 + a)] , 100]]
0.*10^100 + 4.4428829381583662470158809900606936986146216893756902230853956069564\ 34793099473910575326934764765237 I
But with enough digits, it "sees" it's not on the branch and your formula appears correct:
z = 2  I*10^(1000); a = 1/2; Block[{$MaxExtraPrecision = 1100}, N[ HurwitzLerchPhi[1/z, 1, a]  HurwitzLerchPhi[z, 1, a] + 1/a  (1/z)^a*Pi/Sin[Pi*(1 + a)] , 1100]]
0.*10^2199 + 0.*10^2199 I

