Polylogarithms Class Reference

A class for the polylogarithms. More...

#include <Polylogarithms.h>

Inheritance diagram for Polylogarithms:

Detailed Description

A class for the polylogarithms.

Author

HEPfit Collaboration

Copyright

GNU General Public License

This class handles the polylogarithms \(\mathrm{Li}_2(x)\), \(\mathrm{Li}_2(z)\) and \(\mathrm{Li}_3(x)\), where \(x\) and \(z\) are real and complex variables, respectively.

Definition at line 24 of file Polylogarithms.h.

Public Member Functions
gslpp::complex	Li2 (const double x) const
	The dilogarithm with a real argument, \(\mathrm{Li}_2(x)\). More...
gslpp::complex	Li2 (const gslpp::complex z) const
	The dilogarithm with a complex argument, \(\mathrm{Li}_2(z)\). More...
double	Li3 (const double x) const
	The trilogarithm \(\mathrm{Li}_3(x)\). More...
	Polylogarithms ()
	The default constructor. More...
Public Member Functions inherited from BernoulliNumbers
	BernoulliNumbers ()
	The default constructor. More...

Additional Inherited Members
Protected Attributes inherited from BernoulliNumbers
double	B [19]
	the Bernoulli numbers More...

Constructor & Destructor Documentation

Polylogarithms()

Polylogarithms::Polylogarithms

(

)

The default constructor.

Definition at line 16 of file Polylogarithms.cpp.

{
}

Member Function Documentation

Li2() [1/2]

gslpp::complex Polylogarithms::Li2

(

const double

x

)

const

The dilogarithm with a real argument, \(\mathrm{Li}_2(x)\).

This function calls the GSL function gsl_sf_complex_dilog_xy_e().

Parameters

[in]

x

a real variable.

Returns

\(\mathrm{Li}_2(x)\)

Definition at line 22 of file Polylogarithms.cpp.

{
    gsl_sf_result re, im;
    gsl_sf_complex_dilog_xy_e(x, 0.0, &re, &im);
    return gslpp::complex(re.val, im.val, false);
}

Li2() [2/2]

gslpp::complex Polylogarithms::Li2

(

const gslpp::complex

z

)

const

The dilogarithm with a complex argument, \(\mathrm{Li}_2(z)\).

This function calls the GSL function gsl_sf_complex_dilog_xy_e().

Parameters

[in]

z

a complex variable.

Returns

\(\mathrm{Li}_2(z)\)

Definition at line 29 of file Polylogarithms.cpp.

{
    gsl_sf_result re, im;
    gsl_sf_complex_dilog_xy_e(z.real(), z.imag(), &re, &im);
    return gslpp::complex(re.val, im.val, false);
}

Li3()

double Polylogarithms::Li3

(

const double

x

)

const

The trilogarithm \(\mathrm{Li}_3(x)\).

Parameters

[in]

x

a real variable.

Returns

\(\mathrm{Li}_3(x)\)

Attention

This function is applicable for real \(x \leq 1\).

Definition at line 36 of file Polylogarithms.cpp.

{
    double Li3 = 0.0;
    if (x < 0.0)
        Li3 = -gsl_sf_fermi_dirac_2(log(-x));
    else if (x == 0.0)
        Li3 = 0.0;    
    else if (x > 0.0 && x < 0.5) {
        double log_1mx = log(1.0 - x);
        double lfactorial = 1.0, kfactorial = 1.0;
        for (int l=0; l<19; l++) {
            if (l!=0) lfactorial *= (double)l;
            kfactorial = 1.0;
            for (int k=0; k<19; k++) {
                if (k!=0) kfactorial *= (double)k;            
                Li3 += B[l]*B[k]/((double)l+1.0)/((double)l+(double)k+1.0)
                       /lfactorial/kfactorial 
                       * pow(-log_1mx, (double)l+(double)k+1.0);
            }
        }
    } else if (x == 0.5) {
        double log2 = log(2.0);
        double zeta3 = gsl_sf_zeta_int(3);
        Li3 = (4.0*pow(log2, 3.0) - 2.0*M_PI*M_PI*log2 + 21.0*zeta3)/24.0;
    } else if (x > 0.5 && x < 1.0) {
        double log_x = log(x);
        double S12 = 0.0, lfactorial = 1.0;
        for (int l=0; l<19; l++) {
            if (l!=0) lfactorial *= (double)l;
            S12 += 0.5 * B[l]/((double)l+2.0)/lfactorial 
                   * pow(-log_x, (double)l+2.0);
        }
        Li3 = - S12 - log_x*gsl_sf_dilog(1.0-x) - 0.5*log_x*log_x*log(1.0-x)
              + gsl_sf_zeta_int(2)*log_x + gsl_sf_zeta_int(3);
    } else if (x == 1.0)
        Li3 = gsl_sf_zeta_int(3);
    else
        throw std::runtime_error("Polylogarithms::Li3(): x is out of range!");
    return (Li3);
}

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

• Polylogarithms.h
• Polylogarithms.cpp