Previous Page Next Page Contents

dilog -- the dilogarithm function

Introduction

dilog(x) represents the dilogarithm function int(ln(t)/(1-t), t=1..x).

Call(s)

dilog(x)

Parameters

x - an arithmetical expression

Returns

an arithmetical expression.

Overloadable:

x

Side Effects

When called with a floating point argument, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.

Related Functions

ln, polylog

Details

Example 1

We demonstrate some calls with exact and symbolic input data:

>> dilog(0), dilog(2/3), dilog(sqrt(2)), dilog(1 + I), dilog(x)
           2                                           2
         PI                      1/2                 PI
         ---, dilog(2/3), dilog(2   ), - I CATALAN - ---, dilog(x)
          6                                          48

Floating point values are computed for floating point arguments:

>> dilog(-1.2), dilog(3.4 - 5.6*I)
         2.458586602 - 2.477011851 I, - 2.529187195 + 2.25273709 I

Example 2

Arguments built from integers and rational numbers are rewritten, if they lie in the left half of the complex plane or are of absolute value larger than 1. The following arguments have a negative real part:

>> dilog(-400/3), dilog(-1/2 + I)
        2                           2
      PI                   ln(403/3)
      --- + dilog(3/403) + ---------- - ln(403/3) (I PI + ln(400/3))
       6                       2
      
             2              2
           PI    ln(3/2 - I)
         , --- + ------------ + dilog(6/13 + 4/13 I) -
            6         2
      
         ln(- 1/2 + I) ln(3/2 - I)

The following arguments have an absolute value larger than 1:

>> dilog(31/30), dilog(1 + 2/3*I)
                                2                 2
                       ln(31/30)     ln(1 + 2/3 I)
      - dilog(30/31) - ----------, - -------------- -
                           2               2
      
         dilog(9/13 - 6/13 I)

Example 3

The negative real axis is a branch cut of dilog. A jump of height I*2*PI*ln(1-x) occurs when crossing this cut at the real point x<0:

>> dilog(-1.2), dilog(-1.2 + I/10^100), dilog(-1.2 - I/10^100)
      2.458586602 - 2.477011851 I, 2.458586602 - 2.477011851 I,
      
         2.458586602 + 2.477011851 I

Example 4

The functions diff, float, limit, and series handle expressions involving dilog:

>> diff(dilog(x), x, x, x), float(ln(3 + dilog(sqrt(PI))))
             2 ln(x)        2            1
             -------- + ---------- - ----------, 0.8503829845
                    3            2    2
             (1 - x)    x (1 - x)    x  (1 - x)
>> limit(dilog(x^10 + 1)/x, x = infinity)
                                     0
>> series(dilog(x + 1/x)/x, x = -infinity, 4)
               2        2
             PI    ln(x)                  ln(x)
           - --- - ------               - ----- + 1/4
              6      2      ln(x) + 1       2            / 1  \
           -------------- + --------- + ------------- + O| -- |
                 x              2             3          |  4 |
                               x             x           \ x  /

Background

Changes




Do you have questions or comments?


Copyright © SciFace Software GmbH & Co. KG 2000