zeta函数,q-zeta函数,相伴级数与积分

内容简介

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　　《zeta函数，q-zeta函数，相伴级数与积分》解析zeta函数，q—zeta函数，相伴级数与积分的新定义。《zeta函数，q-zeta函数，相伴级数与积分》对zeta函数和q—Zeta函数和相伴级数与积分做一次彻底修改，扩大和更新版本的系列版本。

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目录

PrefaceAcknowledgements1 Introduction and Preliminaries1．1 Gamma and Beta FunctionsThe Gamma FunctionPochhammer’s Symbol and the Factorial FunctionMultiplication Formulas of Legendre and GaussStirling’s Formula for n！ and its GeneralizationsThe Beta FunctionThe Incomplete Gamma FunctionsThe Incomplete Beta FunctionsThe Error FunctionsThe Bohr—Mollerup Theorem1．2 The Euler—Mascheroni Constant γA Set of Known Integral Representations for γFurther Integral Representations for γFrom an Application of the Residue Calculus1．3 Polygamma FunctionsThe Psi （or Digamma） FunctionIntegral Representation for ψ（z）Gauss’s Formulas for ψ（p／q）Special Values of ψ（z）The Polygamma FunctionsSpecial Values of ψ（n） （z）The Asymptotic Expansion for ψ（2）1．4 The Multiple Gamma FunctionsThe Double Gamma Function F2Integral Formulas Involving the Double Gamma FunctionThe Evaluation of an Integral Involving log G（z）The Multiple Gamma FunctionsThe Triple Gamma Function г31．5 The Gaussian Hypergeometric Function and its GeneralizationThe Gauss Hypergeometric EquationGauss’s Hypergeomeu：ic SeriesThe Hypergeometric Series and Its Analytic ContinuationLinear， Quadratic and Cubic TransformationsHypergeometric Representations of Elementary FunctionsHypergeometric Representations of Other FunctionsThe Confluent Hypergeometric FunctionImportant Properties of Kummer’s Confluent Hypergeometric FunctionThe Generalized （Gauss and Kummer） Hypergeometric FunctionAnalytic Continuation of the Generalized Hypergeometric FunctionFunctions Expressible in Terms of the pFq Function1．6 Stirling Numbers of the First and Second KindStirling Numbers of the First KindStirling Numbers of the Second KindRelationships Among Stirling Numbers of the First and Second Kind and Bernoulli Numbers1．7 Bernoulli， Euler and Genocchi Polynomials and NumbersBernoulli Polynomials and NumbersThe Generalized Bernoulli Polynomials and NumbersEuler Polynomials and NumbersFourier Series Expansions of Bernoulli and Euler PolynomialsRelations Between Bernoulli and Euler PolynomialsThe Generalized Euler Polynomials and NumbersGenocchi Polynomials and Numbers1．8 Apostol—Bernoulli， Apostol—Euler and Apostol—Genocchi Polynomials and NumbersApostol—Bernoulli Polynomials and NumbersApostol—Genocchi Polynomials and NumbersImportant Remarks and ObservationsGeneralizations and Unified Presentations of the Apostol Type Polynomials1．9 Inequalities for the Gamma Function and the Double Gamma FunctionThe Gamma Function and Its RelativesThe Double Gamma FunctionProblems2 The Zeta and Related Functions2．1 Multiple Hurwitz Zeta FunctionsThe Analytic Continuation of ζn（S， a）Relationship between ζn（s．x） and B（a）n（x）The Vardi—Barnes Multiple Gamma Functions2．2 The Hurwitz （or Generalized） Zeta FunctionHurwitz’s Formula for ζ（s， a）Hermite’s Formula for ζ（s， a）Further Integral Representations for ζ（s， a）Some Applications of the Derivative Formula （17）Another Form for г2（a）2．3 The Riemann Zeta FunctionRiemann’s Functional Equation for ζ（s）Relationship between ζ（s） and the MathematicalConstants B and CIntegral Representations for ζ（s）A Summation Identity for ζ（n）2．4 PolylogarithmFunctionsThe Dilogarithm FunctionClausen’s Integral （or Function）The Trilogarithm FunctionThe Polylogarithm FunctionsThe Log—Sine Integrals2．5 Hurwitz—Lerch Zeta FunctionsThe Taylor Series Expansion of the Lipschitz—LerchTranscendent L（x， s， a）Evaluation of L（x， —n， a）2．6 Generalizations of the Hurwitz—Lerch Zeta Function2．7 Analytic Continuations of Multiple Zeta FunctionsGeneralized Functions of Gel’fand and ShilovEuler—Maclaurin Summation FormulaProblems3 Series Involving Zeta Functions3．1 Historicallntroduction3．2 Use of the Binomial TheoremApplications of Theorems 3．1 and 3．23．3 Use of Generating FunctionsSeries Involving Polygamma FunctionsSeries Involving Polylogarithm Functions3．4 Use of Multiple Gamma FunctionsEvaluation by Using the Gamma FunctionEvaluation in Terms of Catalan’s Constant GFurther Evaluation by Using the Triple Gamma FunctionApplications of Corollary 3．33．5 Use of Hypergeometric IdentitiesSeries Derivable from Gauss’s Summation Formula 1．4（7）Series Derivable from Kummer’s Formula （3）Series Derivable from Other Hypergeometric Summation FormulasFurther Summation Formulas Related to Generalized Harmonic Numbers3．6 Other Methods and their ApplicationsThe Weierstrass Canonical Product Form for the Gamma FunctionEvaluation by Using Infinite ProductsHigher—Order Derivatives of the Gamma Function3．7 Applications of Series Involving the Zeta FunctionThe Multiple Gamma FunctionsMathieu SeriesProblems4 Evaluations and Series Representations4．1 Evaluation of ζ（2n）The General Case of ζ（2n）4．2 Rapidly Convergent Series for ζ（2n + 1）Remarks and Observations4．3 Further Series Representations4．4 ComputationaIResultsProblems5 Determinants of the Laplacians5．1 The n—Dimensional Problem5．2 Computations Using the Simple and Multiple Gamma FunctionsFactorizations Into Simple and Multiple Gamma FunctionsEvaluations of det’△n （n=1， 2， 3）5．3 Computations Using Series of Zeta Functions5．4 Computations using Zeta Regularized ProductsA Lemma on Zeta Regularized Products and a Main TheoremComputations for small n5．5 Remarks and ObservationsProblems6 q—Extensions of Some Special Functions and Polynomials6．1 q—Shifted Factorials and q—Binormal Coefficients6．2 q—Derivative， q—Antiderivative and Jackson q—lntegralq—Derivativeq—Antiderivative and Jackson q—lntegral6．3 q—Binomial Theorem6．4 q—Gamma Function and q—Beta Functionq—Gamma Functionq—Beta Function6．5 A q—Extension of the Multiple Gamma Functions6．6 q—Bernoulli Numbers and q—Bernoulli Polynomialsq—Stirling Numbers of the Second KindThe Polynomial βk（x）=βk；q（X）6．7 q—Euler Numbers and q—Euler Polynomials6．8 The q—Apostol—Bernoulli Polynomials βk（n） （x；λ） of Order n6．9 The q—Apostol—Euler Polynomials εEk（n）（x；λ） of Order n6．10 A Generalized q—Zeta FunctionAn Auxiliary Function Defining Generalized q—Zeta FunctionApplication of Euler—Maclaurin Summation Formula6．11 Multiple q—Zeta FunctionsAnalytic Continuation of gq and ζqAnalytic Continuation of Multiple Zeta FunctionsSpecial Values of ζq （s1， s2）Problems7 Miscellaneous Results7．1 A Set of Useful Mathematical ConstantsEuler—Mascheroni Constant γSeries Representations for γA Class of Constants Analogous to {Dk}Other Classes of Mathematical Constants7．2 Log—Sine Integrals Involving Series Associated with the Zeta Function and PolylogarithmsAnalogous Log—Sine IntegralsRemarks on Cln （θ） and Gln （θ）Further Remarks and Observations7．3 Applications of the Gamma and Polygamma Functions InvolvingConvolutions of the Rayleigh FunctionsSeries Expressible in Terms of the ψ—FunctionConvolutions of the Rayleigh Functions7．4 Bernoulli and Euler Polynomials at Rational ArgumentsThe Cvijovie—Klinowski Summation FormulasSrivastava’s Shorter Proofs of Theorem 7．3 and Theorem 7．4Formulas Involving the Hurwitz—Lerch Zeta FunctionAn Application of Lerch’s Functional Equation 2．5（29）7．5 Closed—Form Summation of Trigonometric SeriesProblemsBibliography编辑手记

封面

书名:zeta函数,q-zeta函数,相伴级数与积分

作者:(加)斯利瓦斯塔瓦(Srivastava

页数:664

定价:¥88.0

出版社:哈尔滨工业大学出版社

出版日期:2015-08-01

ISBN:9787560355191

PDF电子书大小:144MB 高清扫描完整版