LEADER 04103nam 2200577 450 001 996418186903316 005 20220318144901.0 010 $a3-030-54533-4 024 7 $a10.1007/978-3-030-54533-8 035 $a(CKB)4100000011515553 035 $a(DE-He213)978-3-030-54533-8 035 $a(MiAaPQ)EBC6380776 035 $a(MiAaPQ)EBC6647480 035 $a(Au-PeEL)EBL6380776 035 $a(OCoLC)1202754286 035 $a(Au-PeEL)EBL6647480 035 $a(PPN)258059508 035 $a(EXLCZ)994100000011515553 100 $a20220318d2020 uy 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $anc$2rdacarrier 200 10$aExplorations in complex functions /$fRichard Beals and Roderick S. C. Wong 205 $a1st ed. 2020. 210 1$aCham, Switzerland :$cSpringer,$d[2020] 210 4$d©2020 215 $a1 online resource (XVI, 353 p. 30 illus., 29 illus. in color.) 225 1 $aGraduate Texts in Mathematics,$x0072-5285 ;$v287 311 $a3-030-54532-6 320 $aIncludes bibliographical references and index. 327 $aBasics -- Linear Fractional Transformations -- Hyperbolic geometry -- Harmonic Functions -- Conformal maps and the Riemann mapping theorem -- The Schwarzian derivative -- Riemann surfaces and algebraic curves -- Entire functions -- Value distribution theory -- The gamma and beta functions -- The Riemann zeta function -- L-functions and primes -- The Riemann hypothesis -- Elliptic functions and theta functions -- Jacobi elliptic functions -- Weierstrass elliptic functions -- Automorphic functions and Picard's theorem -- Integral transforms -- Theorems of Phragmén?Lindelöf and Paley?Wiener -- Theorems of Wiener and Lévy; the Wiener?Hopf method -- Tauberian theorems -- Asymptotics and the method of steepest descent -- Complex interpolation and the Riesz?Thorin theorem. 330 $aThis textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book. Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L-functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener?Hopf method. Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout. 410 0$aGraduate Texts in Mathematics,$x0072-5285 ;$v287 606 $aFunctions of complex variables 606 $aMathematical analysis 606 $aFunctions, Special 615 0$aFunctions of complex variables. 615 0$aMathematical analysis. 615 0$aFunctions, Special. 676 $a515 700 $aBeals$b Richard$f1938-$027941 702 $aWong$b Roderick$f1944- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996418186903316 996 $aExplorations in Complex Functions$91887569 997 $aUNISA