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dilog(x) represents the dilogarithm function .
x is a floating-point number, then
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(1+I) = ,
dilog(1-I) = ,
are implemented. For all other arguments,
a symbolic function call.
dilog(x) coincides with
When called with a floating-point 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 floating-point 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
A jump of height 2 π i ln(1
- x) occurs when crossing
this cut at the real point x <
dilog(-1.2), dilog(-1.2 + I/10^100), dilog(-1.2 - I/10^100)
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)
L. Lewin (ed.), “Structural Properties of Polylogarithms”, Mathematical Surveys and Monographs Vol. 37, American Mathematical Society, Providence (1991).
dilog(x) coincides with for |x|
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”: