The dilogarithm is a special function. Everybody knows about the
logarithm. It is the function that turns multiplication into
addition. It has a power series representation as
log(1-x)=-x-x^2/2-x^3/3...-x^n/n...
The dilogarithm has power series representation,
Di(x)=-x-x^2/4-x^3/9...-x^n/n^2...
The dilogarithm is important because is shows up in two places.
The first is in the computation of volume in hyperbolic geometry.
Indeed, the volume of an ideal tetrahedron is expressible in terms
of the dilogarithm. Also the Jones polynomial when computed in terms
of special functions can be seen to be related to the dilogarithm.
The appearance of the dilogarithm in these seemingly unrelated
places is one of the most provocative mathematical discoveries of the
last few years. One of the most active areas of research in topology
at the present time is the attempt to relate the Jones polynomial to
the hyperbolic geometry of knots.