did
Posts:
71
Registered:
9/14/05
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Re: Inversion Lerch Phi
Posted:
Apr 13, 2012 5:11 PM
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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 - 1e-1000 * 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
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