This book is suitable for use in any graduate course on analytical methods and their application to representation theory. Each concept is developed with special emphasis on lucidity and clarity. The book also shows the direct link of Cauchy-Pochhammer theory with the Hadamard-Reisz-Schwartz-Gel'fand et al. regularization. The flaw in earlier works on the Plancheral formula for the universal covering group of SL(2,R) is pointed out and rectified. This topic appears here for the first time in the correct form.Existing treatises are essentially magnum opus of the experts, intended for other experts in the field. This book, on the other hand, is unique insofar as every chapter deals with topics in a way that differs remarkably from traditional treatment. For example, Chapter 3 presents the Cauchy-Pochhammer theory of gamma, beta and zeta function in a form which has not been presented so far in any treatise of classical analysis.
Convergence, Analytic Functions, Complex Integration, Residue Theorem, Cauchy-Pochhammer Theory of Gamma, Beta and Zeta Function; Bargman-Segal Spaces, Elements of the Theory of Generalized Functions; Regularizations and Cauchy's Theory of Analytic Continuation; Gel'fand-Shilov Formulas for Gamma and Beta Function; Lie Group and Invariant Measure; Representations and Unitary Representation; Wigner-Eckart Theorem; SU(2) Group; Elements of SU(3); Gell-Mann Basis and ??-Matrices; Gell-Mann Neeman Octet Model and Mass Formula; Locally Compact Groups: SL(2,R) (SU(1,1)); Principal Exceptional, Positive and Negative Discreet Series and Their Canonical Carrier Spaces; The Clebsch-Gordan Problem: D+ X D+,c; Dc X Dc,e; Group Ring and Invariant Definition of Character; Plancherel Formula as a Completeness Condition of Character; The Group SL(2,C) and Its Unitary Representations; Group Ring and Character; Plancherel Formula; SU(1,1) Content of SL(2,C); Heisenberg-Weyl Group and Its Representations; Coherent-States and Bergman-Segal Spaces; Bargmann's Integral Transform; SU(1,1) Coherent States and Integral Transforms Connecting Well-Known Carrier Spaces of SU(1,1).