|
|
Re: Inversion Lerch Phi
Posted:
Apr 19, 2012 1:09 PM
|
|
Did schrieb:
>
> On 4/12/12 7:15 PM, clicliclic@freenet.de wrote:
> >
> > Did schrieb:
> >>
> >> I am wondering if there is exist an inversion
> >> formula for the Lerch transcendent Phi(x,s,a)
> >> similar to the inversion of the polylogarithms.
> >>
> >> More precisely, I would like to simplify the
> >> expression
> >>
> >> Phi(z,s,a) +/- (-1)^s * Phi(1/z,s,-a) = ???
> >>
> >> at least for s=1. The ??? should be an
> >> expression easier to compute than Phi.
> >>
> >
> > Looks like you want a correct version of Gradshteyn&Ryzhik formula
> > 955.2, taken from Erdelyi. I believe the relation was first derived by
> > Lerch in his Note sur la fonction K(w,x,s) = SUM(e^(2*k*pi*i*x)/(w+k)^s,
> > k, 0, inf) in Acta Math. 11 (1887), 19-24. Lerch probably didn't care
> > much about branch cuts, and Erdelyi then got it definitely wrong.
> >
> > But a correct version exists indeed: it is leading a pieceful life in my
> > vaults. Let's see if somebody can come up with it within a week! And
> > provide a proof within another week!
> >
>
> Thanks Martin, I was actually expecting a reply from you ;-)
>
> Branch cuts do indeed matter to me, specially that I have to
> deal with the possibility |z|>1 (but a is real and not an integer).
> However, for now, I need only s=1 and the plus sign, but it is
> interesting to know the general case, if it exists.
>
> In fact, it's the equivalent formula for the Lerch zeta-function
> I'm the most interested in. (That should be straightforward to
> pass from one to the other, unless I'm missing something about
> the branch cuts.)
>
> Looking forward to see if somebody can come up with it within
> a week!!!
>
One bright Sunday morning I went to church,
And there I met a man named Lerch.
We both did sing in jubilation,
For he did show me a new equation.
- Bruce C. Berndt, 1972.
One week has passed and I don't want to tax your patience any longer.
The following corrected version of Gradshteyn&Ryzhik formula 9.552 is in
my vaults:
Phi(z,s,v) = (-z)^(-v) (2 pi)^(s-1) Gamma(1-s)
[i^(1-s) e^(-i pi v) Phi(e^(-2 pi i v), 1-s, 1/2 + ln(-z)/(2 pi i))
+ i^(s-1) e^(i pi v) Phi(e^(2 pi i v), 1-s, 1/2 - ln(-z)/(2 pi i))]
[either Im(v) < 0, 0 < Re(v) <= 1 or Im(v) >= 0, 0 <= Re(v) < 1].
For s=0, the right-hand side involves something like x*Phi(x, 1, 1/2+y)
- Phi(1/x, 1, 1/2-y) and you have to use G&R formula 9.551 to increase
or decrease the third Phi argument by one. The left-hand side becomes
Phi(z, 0, v) = 1/(1-z), independent of v.
The proof of this formula for complex z,s,v is left to the
sci.math.symbolic public, however.
Martin.
PS: The Maple documentation for LerchPhi(z,s,v) doesn't explicitly say
whether the branch cut for z>1 is to the right or to the left of the
real axis.
<http://www.maplesoft.com/support/help/Maple/view.aspx%3Fpath%3DLerchPhi>
Here are some Derive results for my implementation on the real axis:
VECTOR([z, lerch(z, 1, 1/2)], z, 0, 2, 1/4)
[[0, 2],
0.25, 2.197224577],
0.5, 2.492900959],
0.75, 3.041383984],
1, ?],
1.25, 2.582453645 - 2.809925892*#i],
1.5, 1.87176262 - 2.56509966*#i],
1.75, 1.491669994 - 2.374820823*#i],
2, 1.246450479 - 2.221441469*#i]]
|
|
