Dilogarithm function
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dilog(x
)
dilog(x)
represents the dilogarithm function
.
If x
is a floatingpoint number, then dilog(x)
returns
the numerical value of the dilogarithm function. The special values:
dilog(1)
=
,
dilog(0)
=
,
dilog(1/2)
=
,
dilog(1)
= 0,
dilog(2)
=
,
dilog(I)
=
,
dilog(I)
=
,
dilog(1+I)
=
,
dilog(1I)
=
,
dilog(infinity)
= infinity
are implemented. For all other arguments, dilog
returns
a symbolic function call.
Functional identities are used to rewrite the result for exact
numerical arguments of 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, 1x)
.
When called with a floatingpoint argument, the function is
sensitive to the environment variable DIGITS
which determines
the numerical working precision.
We demonstrate some calls with exact and symbolic input data:
dilog(0), dilog(2/3), dilog(sqrt(2)), dilog(1 + I), dilog(x)
Floating point values are computed for floatingpoint arguments:
dilog(1.2), dilog(3.4  5.6*I)
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)
The following arguments have an absolute value larger than 1:
dilog(31/30), dilog(1 + 2/3*I)
The negative real axis is a branch cut of dilog
.
A jump of height 2 π i 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)
The functions diff
, float
, limit
, and series
handle expressions
involving dilog
:
diff(dilog(x), x, x, x), float(ln(3 + dilog(sqrt(PI))))
limit(dilog(x^10 + 1)/x, x = infinity)
series(dilog(x + 1/x)/x, x = infinity, 3)

Arithmetical expression.
x
dilog(x)
coincides with
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":
.
L. Lewin (ed.), "Structural Properties of Polylogarithms", Mathematical Surveys and Monographs Vol. 37, American Mathematical Society, Providence (1991).