Geometry of Complex Numbers

"Thisbookshouldbeineverylibrary，andeveryexpertinclassicalfunctiontheoryshouldbefamiliarwiththismaterial.Theauthorhasperformedadistinctservicebymakingthismaterialsoconvenientlyaccessibleinasinglebook."—MathematicalReview
Sinceitsinitialpublicationin1962，ProfessorSchwerdtfeger'silluminatingbookhasbeenwidelypraisedforgeneratingadeeperunderstandingofthegeometricaltheoryofanalyticfunctionsaswellasoftheconnectionsbetweendifferentbranchesofgeometry.Itsfocusliesintheintersectionofgeometry，analysis，andalgebra，withtheexpositiongenerallytakingplaceonamoderatelyadvancedlevel.Muchemphasis，however，hasbeengiventothecarefulexpositionofdetailsandtothedevelopmentofanadequatealgebraictechnique.
Inthreebroadchapters，theauthorclearlyandelegantlyapproacheshissubject.Thefirstchapter，AnalyticGeometryofCircles，treatssuchtopicsasrepresentationofcirclesbyHermitianmatrices，inversion，stereographicprojection，andthecrossratio.ThesecondchapterconsidersindepththeMoebiustransformation：itselementaryproperties，realone-dimensionalprojectivities，similarityandclassificationofvariouskinds，anti-homographies，iteration，andgeometricalcharacterization.Thefinalchapter，Two-DimensionalNon-EuclideanGeometries，discussessubgroupsofMoebiustransformations，thegeometryofatransformationgroup，hyperbolicgeometry，andsphericalandellipticgeometry.ForthisDoveredition，ProfessorSchwerdtfegerhasaddedfournewappendicesandasupplementarybibliography.
Advancedundergraduateswhopossessaworkingknowledgeofthealgebraofcomplexnumbersandoftheelementsofanalyticalgeometryandlinearalgebrawillgreatlyprofitfromreadingthisbook.Itwillalsoproveastimulatingandthought-provokingbooktomathematicsprofessorsandteachers.