Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis - often with different definitions and under different names. Among the synonyms are Laplace exponent' instead of Bernstein function, and complete Bernstein functions are sometimes called Pick functions', Nevanlinna functions' or operator monotone functions'.

This monograph - now in its second revised and extended edition - offers a self-contained and unified approach to Bernstein functions and closely related function...