Abook'sapoLogy
Indexofnotation
1Reminders:convergenceofsequencesandseries
1.1Theproblemoflimitsinphysics
1.1.aTwoparadoxesinvolvingkineticenergy
1.1.bRomeo,Juliet,andviscousfluids
1.1.cPotentialwallinquantummechanics
1.1.dSemi-infinitefilterbehavingaswaveguide
1.2Sequences
1.2.aSequencesinanormedvectorspace
1.2.bCauchysequences
1.2.cThefixedpointtheorem
1.2.dDoublesequences
1.2.eSequentialdefinitionofthelimitofafunction
1.2.fSequencesoffunctions
1.3Series
1.3.aSeriesinanormedvectorspace
1.3.bDoublyinfiniteseries
1.3.cConvergenceofadoubleseries
1.3.dConditionallyconvergentseries,absolutelyconvergentseries
1.3.eSeriesoffunctions
1.4Powerseries,analyticfunctions
1.4.aTaylorformulas
1.4.bSomenumericalillustrations
1.4.cRadiusofconvergenceofapowerseries
1.4.dAnalyticfunctions
1.5Aquicklookatasymptoticanddivergentseries
1.5.aAsymptoticseries
1.5.bDivergentseriesandasymptoticexpansions
Exercises
Problem
Solutions
2MeasurethearyandtheLebesgueintegral
2.1TheintegralaccordingtoMr.Riemann
2.1.aRiemannsums
2.1.bLimitationsofRiemann'sdefinition
2.2TheintegralaccordingtoMr.Lebesgue
2.2.aPrincipleofthemethod
2.2.bBorelsubsets
2.2.cLebesguemeasure
2.2.dTheLebesgue-algebra
2.2.eNegligiblesets
2.2.fLebesguemeasureonRn
2.2.gDefinitionoftheLebesgueintegral
2.2.hFunctionszeroalmosteverywhere,spaceL1
2.2.1Andtoday?
Exercises
Solutions
3Integralcalculus
3.1Integrabilityinpractice
3.1.aStandardfunctions
3.l.bComparisontheorems
3.2Exchangingintegralsandlimitsorseries
3.3Integralswithparameters
3.3.aContinuityoffunctionsdefinedbyintegrals
3.3.bDifferentiatingundertheintegralsign
3.3.cCaseofparametersappearingintheintegrationrange
3.4Doubleandmultipleintegrals
3.5Changeofvariables
Exercises
Solutions
4ComplexAnalysisⅠ
4.1Holomorphicfunctions
4.1.aDefinitions
4.2Cauchy'stheorem
4.3Propertiesofholomorphicfunctions
4.4Singularitiesofafunction
4.5Laurentseries
……
5ComplexAnalysisⅡ
6Conformalmaps
7DistributionsⅠ
8DistributionsII
9Hilbertspaces,Fourierseries
10Fouriertransformoffunctions
11Fouriertransformofdistributions
12TheLaplacetransform
13PhysicalapplicationsoftheFouriertransform
14Bras,kets,andallthatsortofthing
15Greenfunctions
16Tensors
17Differentialforms
18Groupsandgrouprepresentations
19Introductiontoprobabilitytheory
20Randomvariables
21Convergenceofrandomvariables:centrallimittheorem
Appendices
Tables