05664nam 2200637 a 450 991046306140332120200520144314.00-88385-970-X(CKB)2670000000386409(EBL)3330358(SSID)ssj0000667044(PQKBManifestationID)11379017(PQKBTitleCode)TC0000667044(PQKBWorkID)10674009(PQKB)10582971(UkCbUP)CR9780883859704(MiAaPQ)EBC3330358(Au-PeEL)EBL3330358(CaPaEBR)ebr10722469(OCoLC)817963747(EXLCZ)99267000000038640920020129d2002 uy 0engurcn|||||||||txtccrInequalities from complex analysis[electronic resource] /John P. D'Angelo[Washington, D.C.] Mathematical Association of Americac20021 online resource (280 p.)The Carus mathematical monographs ;no. 28Description based upon print version of record.0-88385-033-8 Includes bibliographical references (p. 257-259) and index.""Cover ""; ""Title ""; ""Contents""; ""Preface""; ""I. Complex Numbers""; ""I.1 The real number system""; ""I.2 Definition of the complex number field""; ""I.3 Elementary complex geometry""; ""I.4 Alternative definitions of the complex numbers""; ""I.4.1 Using matrices""; ""I.4.2 Using polynomials""; ""I.5 Completeness""; ""I.6 Convergence for power series""; ""I.7 Trigonometry""; ""I.8 Roots of unity""; ""I.9 Summary""; ""II. Complex Euclidean Spaces and Hilbert Spaces""; ""II.1 Hermitian inner products""; ""II.2 Orthogonality, projections and closed subspaces""""II.3 Orthonormal expansion""""II.4 The polarization identity""; ""II.5 Generating functions and orthonormal systems""; ""III. Complex Analysis in Several Variables""; ""III.1 Holomorphic functions""; ""III.2 Some calculus""; ""III.3 The Bergman kernel function""; ""IV. Linear Transformations and Positivity Conditions""; ""IV.1 Adjoints and Hermitian forms""; ""IV.2 Solving linear equations""; ""IV.3 Linearization""; ""IV.4 Eigenvalues and the spectral theorem in finite dimensions""; ""IV.5 Positive definite linear transformations in finite dimensions""; ""IV.6 Hilbert's inequality""""IV.7 Additional inequalities from Fourier analysis""""V. Compact and Integral Operators""; ""V.1 Convergence properties for bounded linear transformations""; ""V.2 Compact operators on Hilbert space""; ""V.3 The spectral theorem for compact Hermitian operators""; ""V.4 Integral operators""; ""V.5 A glimpse at singular integral operators""; ""VI. Positivity Conditions for Real-valued Functions""; ""VI.1 Real variables analogues""; ""VI.2 Real-valued polynomials on C^n""; ""VI.3 Squared norms and quotients of squared norms""; ""VI.4 Plurisubharmonic functions""""VI.5 Positivity conditions for polynomials""""VII. Stabilization and Applications""; ""VII.1 Stabilization for positive bihomogeneous polynomials""; ""VII.2 Positivity everywhere""; ""VII.3 Positivity on the unit sphere""; ""VII.4 Applications to proper holomorphic mappings between balls""; ""VII.5 Positivity on zero sets""; ""VII.6 Proof of stabilization""; ""VIII. Afterword""; ""APPENDIX A""; ""A.1 Algebra""; ""A.2 Analysis""; ""Bibliography""; ""Index""Inequalities from Complex Analysis is a careful, friendly exposition of inequalities and positivity conditions for various mathematical objects arising in complex analysis. The author begins by defining the complex number field, and then discusses enough mathematical analysis to reach recently published research on positivity conditions for functions of several complex variables. The development culminates in complete proofs of a stabilization theorem relating two natural positivity conditions for real-valued polynomials of several complex variables. The reader will also encounter the Bergman kernel function, Fourier series, Hermitian linear algebra, the spectral theorem for compact Hermitian operators, plurisubharmonic functions, and some delightful inequalities. Numerous examples, exercises, and discussions of geometric reasoning appear along the way. Undergraduate mathematics majors who have seen elementary real analysis can easily read the first five chapters of this book, and second year graduate students in mathematics can read the entire text. Some physicists and engineers may also find the topics and discussions useful. The inequalities and positivity conditions herein form the foundation for a small but beautiful part of complex analysis. John P. D'Angelo was the 1999 winner of the Bergman Prize; he was cited for several important contributions to complex analysis, including his work on degenerate Levi forms and points of finite type, as well as work, some joint with David Catlin, on positivity conditions in complex analysisCarus mathematical monographs ;no. 28.Functions of complex variablesInequalities (Mathematics)Mathematical analysisElectronic books.Functions of complex variables.Inequalities (Mathematics)Mathematical analysis.515/.9D'Angelo John P60384MiAaPQMiAaPQMiAaPQBOOK9910463061403321Inequalities from complex analysis1107681UNINA