## 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

ZetaRegularization of | 3 |

Riemann and Hurwitz Zeta Functions | 4 |

References | 20 |

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 |

Applications to Effective Lagrangians for Constant Background Fields | 63 |

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

References | 169 |

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

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

References | 206 |

FiniteTemperature Effects with NonVanishing Chemical Potential | 220 |

Applications to KaluzaKlein Models | 226 |

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

Zeta Functions vs HeatKernel Regularization | 70 |

References | 78 |

Energy | 84 |

ZetaRegularization of Lapla | 107 |

Casimir Energy for Spherical and Cylindrical Universes | 119 |

References | 131 |

Path Integral Techniques and ZetaFunction Regularization of | 143 |

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

References | 257 |

The OneLoop Effective Action for 2D Induced Gravity on a Com | 263 |

282 | |

The SeeleyGilkey Coefficients | 288 |

References | 316 |

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### Common terms and phrases

action analytic continuation appears application associated asymptotic behaviour bosonic boundary boundary conditions calculation Casimir energy classical closed coefficients compact complex computation consider constant contribution convergence corresponding defined definition density dependence derivative determinant dimensions discussion divergences effective effective action eigenvalues energy equation evaluation example exists expansion expression fact factor field Finally finite force formula free energy gauge given gives heat kernel hyperbolic infinite integral interesting introduced invariant known Lett limit manifold Math means method metric namely normal Note Nucl obtained one-loop operator p-brane particular performed Phys plates pole positive possible potential present problem procedure quantity quantum regularization representation residue respectively result Riemann space spacetime string surface taking temperature theory transform vacuum energy vanish zeta function