## Zeta Regularization Techniques with ApplicationsThis book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which appear here, in book format, for the first time are presented. On the other hand, the authors show explicitly how to make use of such formulas and techniques in practical applications to physical problems of very different nature. Virtually all types of zeta functions are dealt with in the book. |

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### Contents

RIEMANN AND RELATED ZETA FUNCTIONS | 1 |

ZetaFunction Regularization of Determinants | 9 |

Uses of Zeta Function in Integration | 15 |

ZETAFUNCTION REGULARIZATION OF SUMS OVER | 23 |

Regularization of Double Series of the Type EpsteinHurwitz | 33 |

Multiple Zeta Functions with Arbitrary Exponents | 43 |

Some Remarks on the Use of the ZetaFunction Regularization | 49 |

ZETA FUNCTION REGULARIZATION GENERALIZED | 57 |

Casimir Energy for Spherical and Cylindrical Universes | 119 |

Path Integral Techniques and ZetaFunction Regularization of | 143 |

Oneloop Effective Action for Scalar Fields on KaluzaKlein Space | 162 |

References | 169 |

124 | 171 |

Path Integral Approach for a Class of Quantum pBrane Models | 183 |

The Casimir Energy for pBranes Compactified on Constant Curva | 197 |

References | 206 |

Applications to Effective Lagrangians for Constant Background Fields | 63 |

Zeta Functions vs HeatKernel Regularization | 70 |

References | 78 |

Definition of the Casimir Energy and its Relation to the Vacuum | 84 |

ZetaRegularization of | 87 |

9 | 104 |

References | 107 |

12 | 111 |

28 | 117 |

FiniteTemperature Effects with NonVanishing Chemical Potential | 220 |

Applications to KaluzaKlein Models | 226 |

Free Energy for the Open Superstring Theory and Laurent Series | 235 |

Thermodynamics of the Open Bosonic String in an External Mag | 248 |

References | 257 |

282 | |

HEATKERNEL AND ZETAFUNCTION REGULAR | 311 |

References | 316 |

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### Common terms and phrases

analytic continuation appears application associated asymptotic bosonic boundary boundary conditions calculation Casimir energy classical closed coefficients compact compute consider constant contribution corresponding defined density dependence derivative determinant dimensions discussion divergence effective action equation evaluation example exists expansion expression fact factor field Finally finite force formalism formula free energy gauge geometry given gives hyperbolic integral introduce invariant known leads Lett limit manifold Math means method metric namely Note Nucl obtained one-loop operator p-brane parameter particular performed Phys plates pole possible potential present problem procedure quantity quantum regularization representation respectively result Riemann scalar field space spacetime string surface taking temperature theory topology transform vacuum energy vanishing zero modes zeta function zeta-function