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This fully revised new edition of a trusted guide to these powerful tools in perturbation theory covers every detail, especially the trickier mathematics. It features new formulae and a wealth of applications including cutting-edge cosmological functions.

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Zeta-function regularization is a powerful method in perturbation theory, and this book is a comprehensive guide for the student of this subject. Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the procedure works in practice, for example in the Casimir effect, gravity and string theory, high-temperature phase transition, topological symmetry breaking, and non-commutative spacetime. The formulae, some of which are new, can be directly applied in creating physically meaningful, accurate numerical calculations. The book acts both as a basic introduction and a collection of exercises for those who want to apply this regularization procedure in practice.

Thoroughly revised, updated and expanded, this new edition includes novel, explicit formulas on the general quadratic, the Chowla-Selberg series case, an interplay with the Hadamard calculus, and also features a fresh chapter on recent cosmological applications, including the calculation of the vacuum energy fluctuations at large scale in braneworld and other models.

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Introduction and Outlook.- Mathematical Formulas Involving the Different Zeta Functions.- A Treatment of the Non-Polynomial Contributions: Application to Calculate Partition Functions of Strings and Membranes.- Analytical and Numerical Study of Inhomogeneous Epstein and Epstein-Hurwitz Zeta Functions.- Physical Application: the Casimir Effect.- Five Physical Applications of The Inhomogeneous Generalized Epstein-Hurwitz Zeta Functions.- Miscellaneous Applications Combinig Zeta With Other Regularization Procedures.- Applications to Gravity, Strings and P-Branes.- Eleventh Application: Topological Symmetry Breaking in Self-Interacting Theories.- Twelth Application: Cosmology and The Quantum-Vacuum.- References.- Index.

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