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Re: Inversion Lerch Phi
Posted:
Apr 13, 2012 3:26 PM
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On 13.04.2012 20:57, Did wrote:
> On 4/13/12 8:28 PM, Axel Vogt wrote:
>> On 13.04.2012 19:54, Did wrote:
>>> On 4/13/12 7:37 PM, Axel Vogt wrote:
>>>> LerchPhi(1/z,1,-a) =
>>>> LerchPhi(z,1,a) - 1/a + (-1/z)^a*Pi/sin(Pi*(1+a));
>>>
>>> Good news. However, this formula appears incorrect at
>>> least for real z>1 (checked numerically with Maple 13).
>>
>> Any explicit example?
>
> Maple program:
>
> restart: Digits:=100: z:=2: a:=1/2:
> evalf( LerchPhi(1/z,1,-a) - LerchPhi(z,1,a) + 1/a - (-1/z)^a*Pi/sin(Pi*(1+a)) );
> 0.+4.442882938158366247015880990060693698614621689375690223085395606956434793099473910575326934764765238*I;
>
> Some numerical tests seem to indicate that your formula
> is valid everywhere except on the branch z>1.
> Thanks for your efforts.
> Did
This is sqrt(2)*Pi* I.
I am not too much used to LerchPhi, but for s=1 and reading it
as 2F1 the branch cut is real axis beyond 1. And z=2 is in it.
Now Maple uses the convention "clockwise around branch point"
to continue into the branch cut.
You can use z = 2 +- 1e-1000*I for test (confirming the formula)
For the given example I _guess_ that the Numerics and the
above rule does not fit together.
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