Public Member Functions | Protected Attributes | List of all members
o2scl::polylog Class Reference

Polylogarithms (approximate) $ Li_n(x)$. More...

#include <polylog.h>

Public Member Functions

double li0 (double x)
 0-th order polylogarithm = $ x/(1-x)$
 
double li1 (double x)
 1-st order polylogarithm = $ -\ln(1-x) $
 
double li2 (double x)
 2-nd order polylogarithm
 
double li3 (double x)
 3-rd order polylogarithm
 
double li4 (double x)
 4-th order polylogarithm
 
double li5 (double x)
 5-th order polylogarithm
 
double li6 (double x)
 6-th order polylogarithm
 

Protected Attributes

double * arg
 
double * two
 
double * three
 
double * four
 
double * five
 
double * six
 
double li2neg1
 
double li4neg1
 
double li6neg1
 

Detailed Description

This class is experimental.

This gives an approximation to the polylogarithm functions.

Only works at present for $n=0,1,...,6$. Uses GSL library for n=2.

Uses linear interpolation for $-1<x<0$ and a series expansion for $x<-1$

Idea for Future:
  • Give error estimate?
  • Improve accuracy?
  • Use more sophisticated interpolation?
  • Add the series $Li(n,x)=x+2^{-n} x^2+3^{-n} x^3+...$ for $ x \rightarrow 0$?
  • Implement for positive arguments < 1.0
  • Make another polylog class which implements series acceleration?

For reference, there are exact relations

\[ \mathrm{Li}_2 \left(\frac{1}{2}\right) = \frac{1}{12}\left[\pi^2-6\left(\ln 2\right)^2\right] \]

\[ \mathrm{Li}_3 \left(\frac{1}{2}\right) = \frac{1}{24}\left[ 4\left(\ln 2\right)^3 - 2 \pi^2 \ln 2 + 21 \zeta (3) \right] \]

\[ \mathrm{Li}_{-1} (x) = \frac{x}{\left(1-x\right)^2} \]

\[ \mathrm{Li}_{-2} (x) = \frac{x\left(x+1\right)}{\left(1-x\right)^3} \]

Definition at line 81 of file polylog.h.


The documentation for this class was generated from the following file:

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