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This text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided. A self-contained and unified approach to the topic With applications to various fields of mathematics, such as probability theory, potential theory, operator theory, integral equations, functional calculi and complex analysis With an extensive list of complete Bernstein functions. Additional material and corrections can be found on the authors' website.

Analytic functions --- Monotonic functions --- Functions, Monotonic --- Functions of real variables --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Bernstein Functions. --- Complete Bernstein Functions. --- Completely Monotone Function. --- Laplace Transform. --- Stieltjes Function/Transform. --- Fonctions analytiques --- Fonctions monotones

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Functional analysis --- Monotonic functions --- Maxima and minima --- Duality theory (Mathematics) --- Monotone operators --- Calculus --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Dualiteit [Theorie van de ] (Wiskunde) --- Dualité [Théorie de la ] (Mathématiques) --- Fonctions monotones --- Functies [Monotoon ] --- Functions [Monotonic ] --- Mathematics duality theory --- Maxima en minima --- Maxima et minima --- Monotone functies --- Monotone operatoren --- Opérateurs monotones --- Theorie van de dualiteit (Wiskunde) --- Théorie de la dualité (Mathématiques) --- Opérateurs monotones --- Dualité, Principe de (Mathématiques) --- Functional analysis. --- Operator theory. --- System theory. --- Calculus of variations. --- Functional Analysis. --- Operator Theory. --- Systems Theory, Control. --- Calculus of Variations and Optimal Control; Optimization. --- Isoperimetrical problems --- Variations, Calculus of --- Systems, Theory of --- Systems science --- Science --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Philosophy --- Monotone operators. --- Algebra --- Mathematical analysis --- Topology --- Operator theory