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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 classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.

Analytic functions. --- Monotonic functions. --- Quasianalytic functions. --- Functions, Quasianalytic --- Quasi-analytic functions --- Quasientire functions in the sense of Bernstein --- Analytic functions --- Functions, Monotonic --- Functions of real variables --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Bernstein Function. --- Monotone Function. --- Probability Measure. --- Semigroup. --- Theory.