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Topic: Inversion Lerch Phi
Replies: 38   Last Post: May 27, 2012 2:36 PM

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 Axel Vogt Posts: 992 Registered: 5/5/07
Re: Inversion Lerch Phi
Posted: Apr 13, 2012 3:26 PM

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.
> 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.

Date Subject Author
4/12/12 did
4/12/12 clicliclic@freenet.de
4/12/12 did
4/19/12 clicliclic@freenet.de
4/26/12 clicliclic@freenet.de
4/26/12 clicliclic@freenet.de
4/26/12 clicliclic@freenet.de
4/26/12 did
4/27/12 clicliclic@freenet.de
5/4/12 clicliclic@freenet.de
5/5/12 did
5/5/12 clicliclic@freenet.de
5/27/12 clicliclic@freenet.de
5/27/12 Axel Vogt
4/13/12 clicliclic@freenet.de
4/13/12 Axel Vogt
4/13/12 did
4/13/12 Axel Vogt
4/13/12 did
4/13/12 did
4/13/12 Axel Vogt
4/13/12 Axel Vogt
4/13/12 did
4/13/12 Axel Vogt
4/13/12 did
4/13/12 Axel Vogt
4/14/12 clicliclic@freenet.de
4/14/12 did
4/14/12 did
4/15/12 clicliclic@freenet.de
4/15/12 did
4/15/12 Axel Vogt
4/15/12 did
4/15/12 Axel Vogt
4/16/12 clicliclic@freenet.de
4/16/12 did
4/14/12 Axel Vogt
4/13/12 Axel Vogt
4/16/12 Joe keane