ComplexAnalysis

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic...

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作者简介

Withthissecondvolume,weentertheintriguingworldofcomplexanalysis.Fromthefirsttheoremson,theeleganceandsweepoftheresultsisevident.Thestartingpointisthesimpleideaofextendingafunctioninitiallygivenforrealvaluesoftheargumenttoonethatisdefinedwhentheargumentiscomplex.Fromthere,oneproceedstothemainpropertiesofholomorphicfunctions,whoseproofsaregenerallyshortandquiteilluminating:theCauchytheorems,residues,analyticcontinuation,theargumentprinciple.Withthisbackground,thereaderisreadytolearnawealthofadditionalmaterialconnectingthesubjectwithotherareasofmathematics:theFouriertransformtreatedbycontourintegration,thezetafunctionandtheprimenumbertheorem,andanintroductiontoellipticfunctionsculminatingintheirapplicationtocombinatoricsandnumbertheory.Thoroughlydevelopingasubjectwithmanyramifications,whilestrikingacarefulbalancebetweenconceptualinsightsandthetechnicalunderpinningsofrigorousanalysis,"ComplexAnalysis"willbewelcomedbystudentsofmathematics,physics,engineeringandothersciences."ThePrincetonLecturesinAnalysis"representsasustainedefforttointroducethecoreareasofmathematicalanalysiswhilealsoillustratingtheorganicunitybetweenthem.Numerousexamplesandapplicationsthroughoutitsfourplannedvolumes,ofwhich"ComplexAnalysis"isthesecond,highlightthefar-reachingconsequencesofcertainideasinanalysistootherfieldsofmathematicsandavarietyofsciences.SteinandShakarchimovefromanintroductionaddressing"Fourier"seriesandintegralstoin-depthconsiderationsofcomplexanalysis;measureandintegrationtheory,andHilbertspaces;and,finally,furthertopicssuchasfunctionalanalysis,distributionsandelementsofprobabilitytheory.