标题：数论中的模函数和狄利克莱级数
图书信息
出版社: 世界图书出版公司; 第2版 (2009年4月1日)
外文书名: Modular Functions and Dirichlet Series in Number Theory (2nd Edition)
平装: 204页
正文语种: 英语
开本: 24
ISBN: 7510004403, 9787510004407
条形码: 9787510004407
尺寸: 22 x 14.8 x 1.2 cm
重量: 299 g
作者简介
作者：(美国) 阿波斯托尔 (Apostol.T.M.)
内容简介
《数论中的模函数和狄利克莱级数(第2版)》讲述了：This is the second volume of a 2-volume textbook* which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years.The second volume presupposes a background in number theory com-parable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis
目录
Chapter1 Ellipticfunctions
1.1 Introduction
1.2 Doublyperiodicfunctions
1.3 Fundamentalpairsofperiods
1.4 Ellipticfunctions
1.5 Constructionofellipticfunctions
1.6 TheWeierstrassfunction
1.7 TheLaurentexpansionofganeartheorigin
1.8 Differentialequationsatisfiedbyξ
1.9 TheEisensteinseriesandtheinvariantsg2andg3
1.10 Thenumberse1,e2,e3
1.11 ThediscriminantA
1.12 Klein'smodularfunctionJ(τ)
1.13 InvarianceofJunderunimodulartransformations
1.14 TheFourierexpansionsofg2(τ)andg3(τ)
1.15 TheFourierexpansionsof△(τ)andJ(τ)
ExercisesforChapter1
Chapter2 TheModulargroupandmodularfunctions
2.1 M6biustransformations
2.2 Themodulargroup
2.3 Fundamentalregions
2.4 Modularfunctions
2.5 Specialvaluesof
2.6 Modularfunctionsasrationalfunctionsof
2.7 Mappingpropertiesof
2.8 ApplicationtotheinversionproblemforEisensteinseries
2.9 ApplicationtoPicard'stheorem
ExercisesforChapter2
Chapter3 TheDedekindetafunction
3.1 Introduction
3.2 Siegei'sproofofTheorem3.1 
3.3 Infiniteproductrepresentationfor△(τ)
3.4 Thegeneralfunctionalequationforη(τ)
3.5 Iseki'stransformationformula
3.6 DeductionofDedekind'sfunctionalequationfromIseki'sformula
3.7 PropertiesofDedekindsums
3.8 ThereciprocitylawforDedekindsums
3.9 CongruencepropertiesofDedekindsums
3.1 0TheEisensteinseriesG2(τ)
ExercisesforChapter3
Chapter4 Congruencesforthecoefficientsofthemodularfunctionj
4.1 Introduction
4.2 ThesubgroupFo(q)
4.3 FundamentalregionofFo(p)
4.4 FunctionsautomorphicunderthesubgroupFo(p)
4.5 ConstructionoffunctionsbelongingtoFo(p)
4.6 Thebehavioroffpunderthegeneratorsofг
4.7 Thefunction(τ)=△(qτ)/△(τ)
4.8 Theunivalentfunctionφ(τ)
4.9 Invarianceofφ(τ)undertransformationsofг0(q)
4.1 0Thefunctionjpexpressedasapolynomialinφ
ExercisesforChapter4
Chapter5 Rademacher'sseriesforthepartitionfunction
5.1 Introduction
5.2 Theplanoftheproof
5.3 Dedekind'sfunctionalequationexpressedintermsofF
5.4 Fareyfractions
5.5 Fordcircles
5.6 Rademacher'spathofintegration
5.7 Rademacher'sconvergentseriesforp(n)
ExercisesforChapter5
Chapter6 Modularformswithmultiplicativecoefficients
6.1 Introduction
6.2 Modularformsofweightk
6.3 Theweightformulaforzerosofanentiremodularform
6.4 RepresentationofentireformsintermsofG4andG6
6.5 ThelinearspaceMkandthesubspaceMk.o
6.6 Classificationofentireformsintermsoftheirzeros
6.7 TheHeckeoperatorsTn
6.8 Transformationsofordern
6.9 BehaviorofTnfunderthemodulargroup
6.10 MultiplicativepropertyofHeckeoperators
6.11 EigenfunctionsofHeckeoperators
6.12 Propertiesofsimultaneouseigenforms
6.13 Examplesofnormalizedsimultaneouseigenforms
6.14 RemarksonexistenceofsimultaneouseigenformsinM2k.0
6.15 EstimatesfortheFouriercoefficientsofentireforms
6.16 ModularformsandDirichletseries
Exerci