Re: Polylog equations
- To: mathgroup at smc.vnet.net
- Subject: [mg84990] Re: Polylog equations
- From: sashap <pavlyk at gmail.com>
- Date: Tue, 22 Jan 2008 01:52:29 -0500 (EST)
- References: <fn21oa$in5$1@smc.vnet.net>
On Jan 21, 6:04 am, Valeri Astanoff <astan... at gmail.com> wrote:
> Good day,
>
> Given these polylog equations:
>
> In[1]:= Assuming[0 < x < 1,
> Solve[PolyLog[3/2,x]==y && PolyLog[5/2,x]==z,z,x]]
>
> Out[1]= {{}}
>
> what is the best way to get z(y)?
>
> Thanks,
>
> V.Astanoff
One way to get an approximation is to use ComposeSeries and
InverseSeries;
f[n_Integer?Positive, x_] :=
Module[{y}, (ComposeSeries[
InverseSeries[Series[PolyLog[3/2, y], {y, 0, n}]],
Series[PolyLog[5/2, y], {y, 0, n}]] // Simplify // Normal) /.
y -> x]
In[49]:= f[2, PolyLog[3/2, x]] - PolyLog[5/2, x] /. x -> x + O[x]^4
Out[49]= SeriesData[x, 0, {-1/8 + 2/(9*Sqrt[3])}, 3, 4, 1]
The series obtained converges for 0<y<1:
Show[ParametricPlot[{PolyLog[3/2, x], PolyLog[5/2, x]}, {x, 0, 1},
PlotStyle -> Directive[Thick, Red]],
Plot[Evaluate[Table[f[n, y], {n, 2, 6}]], {y, 0, PolyLog[3/2, 1]}]]
Oleksandr Pavlyk