In mathematics , various special functions are called a dilogarithm . The classic dilogarithm is a special case of the polylogarithm .
Classic dilogarithm
Values of the classical dilogarithm on the real axis. (The imaginary part is identically zero there.)
The classic Dilogarithm for complex numbers with defined by the power series
-
.
It can be continued through analytical continuation on :
(This must be integrated along a path in .)
Bloch-Wigner dilogarithm
The Bloch-Wigner dilogarithm is defined for by
-
.
It's well-defined and steady, even in .
It is analytical in , in 0 and 1 it has singularities of type .
Rogers dilogarithm
The Rogers dilogarithm is defined by
for .
Another common definition is
-
.
This depends on the former via
together.
One can (discontinuously) continue on completely through and
Elliptical dilogarithm
Let be an elliptic curve defined over . It can be parameterized by
means of a grid using the Weierstrasse schen function
-
mod .
The elliptical dilogarithm is then defined by
-
,
where denotes the Bloch-Wigner dilogarithm.
The elliptical dilogarithm agrees with the value of the L-function except for rational multiples of .
Special values
Classic dilogarithm
For the following numbers, and can be represented in closed form:
-
,
-
,
-
,
-
.
With the sixth root of unity and the Gieseking constant one also has
-
.
Bloch-Wigner dilogarithm
So far, values of the Bloch-Wigner dilogarithm can only be calculated numerically and only a few algebraic relations are known between values of the Bloch-Wigner dilogarithm. An assumption by John Milnor states that :
- the numbers for and are linearly independent over .
Rogers dilogarithm
There are numerous algebraic identities between values of in rational or algebraic arguments. Examples of special values are
-
.
With the sixth root of unity and the Gieseking constant one has
Functional equations
Classic dilogarithm
The classical dilogarithm suffices for numerous functional equations, for example
Bloch-Wigner dilogarithm
The Bloch-Wigner dilogarithm is sufficient for identities
and the 5-term relation
Rogers dilogarithm
The Rogers dilogarithm fulfills the relationship
and Abel's functional equation
-
.
For one has
and the 5-term relation
-
,
in particular is a well-defined function on the Bloch group .
See also
Web links
Individual evidence
-
^ K 2 and L-functions of elliptic curves