dilog -- the dilogarithm
function
Introductiondilog(x) represents the dilogarithm
function int(ln(t)/(1-t), t=1..x).
Call(s)dilog(x)
Parametersx |
- | an arithmetical expression |
Returnsan arithmetical expression.
x
Side
EffectsWhen called with a floating point argument, the function is
sensitive to the environment variable DIGITS which determines the numerical
working precision.
Related
Functions
Detailsx is a floating point number, then
dilog(x) returns the numerical value of the
dilogarithm function. The special values:
dilog(-1) = PI^2/4-I*PI*ln(2),
dilog(0) = PI^2/6,
dilog(1/2) = PI^2/12-ln(2)^2/2,
dilog(1) = 0,
dilog(2) = -PI^2/12,
dilog(I) =
PI^2/16-I*CATALAN-I*PI*ln(2)/4,
dilog(-I)=
PI^2/16+I*CATALAN+I*PI*ln(2)/4,
dilog(1+I) = -PI^2/48-I*CATALAN,
dilog(1-I) = -PI^2/48+I*CATALAN,
dilog(infinity) = -infinity
are implemented. For all other arguments, dilog returns
a symbolic function call.
Type::Numeric that have a negative
real part or are of absolute value larger than 1. Cf.
example 2.dilog(x) coincides with polylog(2, 1-x).
Example
1We 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
2Arguments 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
3The 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
4The 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 /
Backgrounddilog(x) coincides with
sum((1-x)^k/k^2,k=1..infinity) for |x|<1.dilog has a branch cut along the negative real axis.
The value at a point x on the cut coincides with the limit
``from above'':
dilog(x)
=limit(dilog(x+eps*I), eps=0, Right)
=limit(dilog(x+eps*I), eps=0, Left) - 2*PI*I*ln(1-x).
series attribute was added.