Re: Polylog equations
- To: mathgroup at smc.vnet.net
- Subject: [mg85136] Re: Polylog equations
- From: Valeri Astanoff <astanoff at gmail.com>
- Date: Wed, 30 Jan 2008 06:01:40 -0500 (EST)
- References: <fn21oa$in5$1@smc.vnet.net> <fne26h$5at$1@smc.vnet.net>
On 26 jan, 02:25, "David W.Cantrell" <DWCantr... at sigmaxi.net> wrote:
> Valeri Astanoff <astan... at gmail.com> wrote:
> > Good day,
>
> > Given these polylog equations:
>
> > In[1]:= Assuming[0 < x < 1,
> > =A0 =A0 =A0 =A0 Solve[PolyLog[3/2,x]==y && PolyLog[5/2,x]==z,z,x=
]]
>
> > Out[1]= {{}}
>
> > what is the best way to get z(y)?
>
> I haven't seen my previous reply or any replies from others appear in the
> newsgroup yet. But Valeri and I have had some correspondence by private
> email. From that, here's some information that may interest others.
>
> According to Valeri, Albert Einstein gave an approximation for z(y) in
> writing about quantum theory of ideal gases: z(y) is approximately
>
> y - 0.1768 y^2 - 0.0034 y^3 - 0.0005 y^4
>
> But those appear to be merely the first four terms of a Maclaurin series,
> which I found easily with Mathematica:
>
> In[7]:= Normal[
> =A0PolyLog[5/2, InverseSeries[Series[PolyLog[3/2, x], {x, 0, 4}], y]]]
>
> Out[7]= y - y^2/(4*Sqrt[2]) + (1/8 - 2/(9*Sqrt[3]))*y^3 +
> =A0 =A0(-(3/32) - 5/(32*Sqrt[2]) + 1/(2*Sqrt[6]))*y^4
>
> In[8]:= N[%]
>
> Out[8]= y - 0.176777 y^2 - 0.00330006 y^3 - 0.000111289 y^4
>
> and so it would seem that, if Einstein had intended to give the first four=
> terms of that Maclaurin series, either his numerical work was not quite
> right or I (or Mathematica) made a mistake.
>
> Valeri also noted that z(Zeta[3/2]) is Zeta[5/2], which is about 1.34148..=