Volkenborn-Integral

The people Born- integral is an integral concept for functions on the p- adic numbers.

Definition

Be

A locally - analytic function of, the ring of p- adic integers, in the completion of the algebraic completion of, the body of the - adic numbers (a function is called locally - analytic if there is a disc centered at each point within which can develop the function as a power series ). The people of Born- integral is then defined by

Formation

The idea of ​​integration of p -adic functions had initially F. Thomas and F. Bruhat. However, the definition of their translation-invariant p- adic integral proved to be too restrictive for analytic and number theoretic purposes.

Arnt People Born developed in his dissertation at the University of Cologne in 1971 which was later named after him -adic generalized integral. With the people Born- integral are all locally analytic functions, such as Laurent series, integrable. Application finds the people Born- integral in the calculation of the so-called generalized Bernoulli numbers and other -adic functions.