Clausen function

In mathematics, the Clausen function is defined by the following integral:

General definition

More generally one defines complex with:

This definition can be continued over the entire complex plane analytically.

Relation to Polylogarithmus

The Clausen function is related to the Polylogarithmus:

Woe relationship

Ernst Kummer and Rogers following result for valid relationship to:

Relationship to the Dirichlet L-functions

For rational values ​​of the function may be regarded as a periodic orbit of an element of a cyclic group. Consequently, can be considered as a simple sum that includes the Hurwitz zeta function. This makes it easy to calculate relationships between certain Dirichlet L-functions.

The Clausen function as a regularization method

The Clausen function can also be viewed as a way to give meaning to the following divergent Fourier series:

Which may be referred to. By integration we obtain:

This result can be generalized by analytic continuation for all negative.

Series expansion

A series expansion for the Clausen function ( for ) is

Is the Riemann zeta function. A faster converging series is

The convergence is ensured that for large quickly converges to 0.

Special values

Some special values ​​are:

Where G is the constant Catalansche.

General:

The Dirichlet beta function.