Dilogarithm

In mathematics, various special functions are called Dilogarithm. The classic Dilogarithm is a special case of Polylogarithmus.

• 6.1 Classic Dilogarithm
• 6.2 Bloch -Wigner Dilogarithm
• 6.3 Rogers Dilogarithm

Classic Dilogarithm

The classic Dilogarithm is for complex numbers defined by the power series

He goes on and on by analytic continuation:

(This has to be integrated along a path in. )

Bloch -Wigner Dilogarithm

The Bloch -Wigner Dilogarithm is defined by

He is well - defined and continuous, even in.

It is analytic in, 0 and 1, it has singularities of type.

Rogers Dilogarithm

Rogers Dilogarithm is defined by

Elliptical Dilogarithm

Be an elliptic curve defined over. By means of the Weierstrass p- function, it can be parameterized by means of a grid

The elliptical Dilogarithm is then defined by

Where the Bloch -Wigner Dilogarithm called.

Special values

Classic Dilogarithm

There are exactly eight numbers, leave for and represent in closed form:

Bloch -Wigner Dilogarithm

Values ​​of the Bloch -Wigner Dilogarithm can so far only be calculated numerically, and you only know a few algebraic relations between values ​​of the Bloch -Wigner Dilogarithm. A conjecture of John Milnor states for:

Rogers Dilogarithm

There are numerous algebraic identities between values ​​of in rational or algebraic arguments. Examples of particular values ​​are

Functional equations

Classic Dilogarithm

The classic Dilogarithm sufficient number of functional equations, for example

Bloch -Wigner Dilogarithm

The Bloch -Wigner Dilogarithm satisfies the identities

And the 5- term ratio

Rogers Dilogarithm

Rogers Dilogarithm satisfies the relation

And Abel's functional equation