Complex analysis, complex variables Books

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Book SynopsisFrom the contents:Preface for the English edition; V.I. Arnold.- Preface.- Introduction.- 1: Groups.- 2: The complex numbers.- 3: Hints, Solutions and Answers.- Appendix. Solvability of equations by explicit formulae; A. Khovanskii.- Bibliography.- Appendix; V.I. Arnold.- Index. Trade ReviewFrom the reviews: "This very special and brilliant text has been written for bright non-specialists in mathematics, but it leads the reader up to topical research problems in the field, and that in a masterly manner. The book is absolutely self-contained, in its own particular fashion, and it is therefore perfectly suited for self-study, ranging from advanced high school to graduate level. No doubt, the thorough and serious working with this outstanding text could turn very beginners into creative almost-experts in the field." (Werner Kleinert, Zentralblatt MATH, Vol. 1065 (16), 2005)Table of ContentsPreface for the English edition; V.I. Arnold. Preface. Introduction. 1: Groups. 1.1. Examples. 1.2. Groups of transformations. 1.3. Groups. 1.4. Cyclic groups. 1.5. Isomorphisms. 1.6. Subgroups. 1.7. Direct product. 1.8. Cosets. Lagrange's theory. 1.9. Internal automorphisms. 1.10. Normal subgroups. 1.11. Quotient groups. 1.12. Commutant. 1.13. Homomorphisms. 1.14. Soluble groups. 1.15. Permutations. 2: The complex numbers. 2.1. Fields and polynomials. 2.2. The field of complex numbers. 2.3. Uniqueness of the field of complex numbers. 2.4. Geometrical descriptions of the field of complex numbers. 2.5. The trigonometric form of the complex numbers. 2.6. Continuity. 2.7. Continuous curves. 2.8. Images of curves: the basic theorem of the algebra of complex numbers. 2.9. The Riemann surface of the function w = SQRTz. 2.10. The Riemann surfaces of more complicated functions. 2.11. Functions representable by radicals. 2.12. Monodromy groups of multi-valued functions. 2.13. Monodromy groups of functions representable by radicals. 2.14. The Abel theorem. 3: Hints, Solutions and Answers. 3.1.Problems of Chapter 1. 3.2. Problems of Chapter 2. Drawings of Riemann surfaces; F. Aicardi. Appendix. Solvability of equations by explicit formulae; A. Khovanskii. A.1. Explicit solvability of equations. A.2. Liouville's theory. A.3. Picard-Vessiot's theory. A.4. Topological obstructions for the representation of functions by quadratures. A.5. S-functions. A.6. Monodromy group. A.7. Obstructions for the representability of functions by quadratures. A.8. Solvability of algebraic equations. A.9. The monodromy pair. A.10. Mapping of the semi-plane to a polygon bounded by arcs of circles. A.11. Topological obstructions for the solvability of differential equations. A.12. Algebraic functions of several variables. A.13. Functions of several complex variables representable by quadratures and generalized quadratures. A.14. SC-germs. A.15. Topological obstruction for the solvability of the holonomic systems of linear differential equations. A.16. Topological obstruction for the solvability of the holonomic systems of linear differential equations. Bibliography. Appendix; V.I. Arnold. Index.

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Book SynopsisPreface to Second Edition.- Preface to First Edition.- Standard Notation and Commonly Used Symbols.- 1 The Fundamental Theorem in Complex Function Theory.- 2 Foundations.- 3 Power Series.- 4 The Cauchy Theory - A Fundamental Theorem.- 5 The Cauchy Theory - Key Consequences.- 6 Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions.- 7 Sequences and Series of Holomorphic Functions.- 8 Conformal Equivalence and Hyperbolic Geometry.- 9 Harmonic Functions.- 10 Zeros of Holomorphic Functions.- Bibliographical Notes.- Bibliography.- Index.Trade ReviewFrom the reviews of the second edition:“The book under review is a second edition of a book by the same authors and with the same title, also published by Springer in 2007. It contains some amount of new information. … The book is carefully written and each chapter has an interesting list of exercises. I found it very useful as am Undergraduate and Graduate Text in Mathematics.” (José M. Ansemil, The European Mathematical Society, euro-math-soc.eu, January, 2014)From the reviews:“The book is written in a clear and easily readable manner, completed with a well cared selection of exercises. The book, which is certainly useful for all specialists and lecturers in complex analysis, I also recommend to beginners in this area.” (Fernando Perez-Gonzalez, Zentralblatt MATH, Vol. 1262, 2013)"This is a fairly conventional text for a first course in complex analysis. It is an interesting mix of the concrete and the abstract, and of the formulaic and the geometric. It has good exercises … . It is nominally a graduate text (it is in Springer’s series of Graduate Texts in Mathematics) … . The book covers all the usual topics for a first course and includes a lot of advanced topics … ." (Allen Stenger, MathDL, February, 2008)"This book is based on the original courses of Complex Analysis that was delivered by the well-known American specialist and lecturer in Analysis, Professor Lipman Bers. … the book is written in a clear and easily readable manner. … this book is useful for all specialists and lecturers in Complex Analysis … and also all specialists who deal with applications of Complex Analysis. I also recommend this book to beginners who want to study Complex Analysis … ." (Peter Zabreiko, Zentralblatt MATH, Vol. 1139 (17), 2008)Table of ContentsPreface to Second Edition.- Preface to First Edition.- Standard Notation and Commonly Used Symbols.- 1 The Fundamental Theorem in Complex Function Theory.- 2 Foundations.- 3 Power Series.- 4 The Cauchy Theory - A Fundamental Theorem.- 5 The Cauchy Theory - Key Consequences.- 6 Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions.- 7 Sequences and Series of Holomorphic Functions.- 8 Conformal Equivalence and Hyperbolic Geometry.- 9 Harmonic Functions.- 10 Zeros of Holomorphic Functions.- Bibliographical Notes.- Bibliography.- Index.

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Book SynopsisPreface.- Chapter 1. Preliminaries.- Chapter 2. Fock Spaces.- Chapter 3. The Berezin Transform and BMO.- Chapter 4. Interpolating and Sampling Sequences.- Chapter 5. Zero Sets for Fock Spaces.- Chapter 6. Toeplitz Operators.- Chapter 7. Small Hankel Operators.- Chapter 8. Hankel Operators.- References.- Index.Trade ReviewFrom the reviews:“Excellent books exist in the literature on the theory of Hardy spaces … but no textbook concerning the theory of Fock spaces has appeared before. The purpose of the author is to fill this gap and provide to any researcher in the field or graduate students the appropriate place to find the results or the bibliographical references needed for their use. … author succeeds with his goal. … a great addition to the literature and in the future will become a classic in the field.” (Jordi Pau, Mathematical Reviews, January, 2013)“This book is intended to provide a convenient reference to Fock spaces. … Each chapter ends with a series of exercises. The material is presented in a pedagogical way. The reference list contains 259 relevant items. This book is well written and it is a good reference for graduate students who are interested in Fock spaces.” (Atsushi Yamamori, Zentralblatt MATH, Vol. 1262, 2013)Table of ContentsPreface.- Chapter 1. Preliminaries.- Chapter 2. Fock Spaces.- Chapter 3. The Berezin Transform and BMO.- Chapter 4. Interpolating and Sampling Sequences.- Chapter 5. Zero Sets for Fock Spaces.- Chapter 6. Toeplitz Operators.- Chapter 7. Small Hankel Operators.- Chapter 8. Hankel Operators.- References.- Index.

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Book SynopsisThis book provides a concise survey of the theory of zero product-determined algebras, which has been developed over the last 15 years. It is divided into three parts. The first part presents the purely algebraic branch of the theory, the second part presents the functional analytic branch, and the third part discusses various applications. The book is intended for researchers and graduate students in ring theory, Banach algebra theory, and nonassociative algebra.Trade Review“This book is about zero product determined algebras and is written in an attractive way. It deals with the introduction and study of this class of algebras. Most of this book is taken from research articles from the last 15 years and is suitable for researchers in this field and students with different backgrounds and can be used for self-study.” (Hoger Ghahramani, Mathematical Reviews, March, 2023)Table of Contents- Part I Algebraic Theory. - Zero Product Determined Nonassociative Algebras. - Zero Product Determined Rings and Algebras. - Zero Lie/Jordan Product Determined Algebras. - Part II Analytic Theory. - Zero Product Determined Nonassociative Banach Algebras. - Zero Product Determined Banach Algebras. - Zero Lie/Jordan Product Determined Banach Algebras. - Part III Applications. - Homomorphisms and Related Maps. - Derivations and Related Maps. - Miscellany.

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Book SynopsisIntegrals and sums are not generally considered for evaluation using complex integration. This book proposes techniques that mainly use complex integration and are quite different from those in the existing texts. Such techniques, ostensibly taught in Complex Analysis courses to undergraduate students who have had two semesters of calculus, are usually limited to a very small set of problems. Few practitioners consider complex integration as a tool for computing difficult integrals. While there are a number of books on the market that provide tutorials on this subject, the existing texts in this field focus on real methods. Accordingly, this book offers an eye-opening experience for computation enthusiasts used to relying on clever substitutions and transformations to evaluate integrals and sums. The book is the result of nine years of providing solutions to difficult calculus problems on forums such as Math Stack Exchange or the author's website, residuetheorem.com. It serves to detail to the enthusiastic mathematics undergraduate, or the physics or engineering graduate student, the art and science of evaluating difficult integrals, sums, and products.Table of Contents1. Review of foundational concepts1.1. Sequences and Series 1.1.1. Sequences of Real Numbers and their Series – sequences, limits, series, convergence, harmonic numbers, summation by parts, change in the order of summation 1.1.2. Power Series and Generating Functions – definitions, radius of convergence, generating function representations of sequences, convolution 1.2. Integrals 1.2.1. Riemann Sums – definition, direct evaluation of certain sums 1.2.2. Fundamental Theorem - definition of definite integral, statement of theorem, verifications 1.2.3. Multiple Integrals – double integrals, conditions for reversal or order of integration 1.3. Evaluation Techniques 1.3.1. Integration by Parts - review 1.3.2. Conversion to Multiple Integrals – “Feynman’s Technique,” replacing a portion of an integrand with an integral representation and reversing the order of integration 1.3.3. Green’s Theorem – review, path integrals and parametrization, Stokes’ Theorem, applications 1.3.4. Partial Fractions review 1.4. Problems 2. Complex Integration 2.1. Analytic Functions 2.1.1. Cauchy-Riemann Conditions – complex functions and their derivatives, defining analytic functions as a direction-independent derivative, harmonic functions 2.1.2. Evaluating Complex Integrals – numerical examples of parametrizations 2.1.3. Path Independence – demonstrate for analytic functions and demonstrate invalidity for nonanalytic integrands 2.2. Cauchy’s Theorems 2.2.1. Winding Numbers – definition in terms of a complex integral 2.2.2. Cauchy’s Integral Theorem – derivation and illustration for a wide variety of integrands and contours 2.2.3. Cauchy’s Theorem – statement, examples, Liouville’s Theorem, Morera’s Theorem 2.3. Useful Results 2.3.1. Taylor Series – review, error analysis in complex plane, convergence 2.3.2. Laurent Series – regions of validity (e.g., annuli), analytic continuation 2.3.3. Argument Principle – derivation for zeroes and poles 2.3.4. Rouche’s Theorem – derivation, illustration for determining poles within integration contours 2.4. Multivalued Functions – branch points, branch cuts, Riemann surfaces 2.5. Problems 3. Evaluation of Real Integrals and Sums 3.1. Preliminary Matters 3.1.1. Poles and Residue Theory – residue definition, residue computation 3.1.2. Essential Singularities – computation of residues of essential singularities 3.1.3. Branch Points – illustration of a unified approach to expressing an integral of a function in terms of its singularities 3.2. Definite Integrals 3.2.1. Integrands Having Both Poles and Branch Points – e.g., integrands featuring logs and exponents less than -1 3.2.2. Integrands Defined Over - insertion of one higher power of log(z) in the integrand, residue backpropagation 3.2.3. Integrands Having Rational Functions of Polynomials and Trigonometric Functions – integration over the unit circle, modifying the unit circle in the presence of singularities, replacing monomial with a branch point in constructing a contour integral 3.2.4. Alternative Contours: Wedges, Rectangles, and Others – reducing the number of singularities in a contour to simplify calculation 3.2.5. Integrands Having Algebraic Functions and the Residue At Infinity – whole new paradigm in evaluating definite integrals with finite limits of an integrand having branch points at the finite limits, defining the residue at infinity, branch point at infinity 3.3. Sums 3.3.1. Complex Integral Representations – selection of integrand and contour to produce sums, demonstration of convergence of complex integral as contour expands to infinity 3.3.2. Examples – rational summands, summands with trigonometric functions 3.4. Problems 4. Cauchy Principal Value 4.1. Integrands Having Poles On the Contour 4.1.1. Definition of a Cauchy Principal Value – definition as a limit, illustration with simple examples 4.1.2. Managing Divergent Terms of a Contour Integral – detailed illustrations of evaluating definite integrals via complex integrals having contributions with divergent terms that cancel 4.2. Analytic Signals and Hilbert Transforms – equivalence of Cauchy-Riemann equations and Hilbert transforms of real and imaginary parts of an analytic function, illustrations of analytic signals having harmonic real and imaginary parts, examples of deriving imaginary parts of analytic function from real part 4.3. Problems 5. Integral Transforms 5.1. Preliminary Matters 5.1.1. The Dirac Delta Function – derivation via self-transform in Hilbert transform integrals, review of properties 5.1.2. A General Discussion of Integral Transforms - integral transforms require a computable inverse to be of any use, conditions under which inverses exist, general format of integral transforms 5.2. The Fourier Transform 5.2.1. Definition and Plancherel’s Theorem – mean square error, and inner product spaces, the Fourier Transform as a Principal Value 5.2.2. Jordan’s Lemma – evaluating Fourier integrals using complex integration, convergence conditions 5.2.3. Parseval’s Theorem – statement, examples of integral evaluations, Fourier series and application of theorem to sums 5.2.4. Convolution Theorem – statement and derivation, applications 5.2.5. Analyticity of the Fourier Transform In the Complex Plane - theorem relating rates of convergence of Fourier transforms and their inverses in the complex plane, strips of convergence, causality 5.2.6. Poisson Sum Formula – derivation, application to computation of error function to machine precision anywhere in the complex plane 5.3. The Laplace Transform 5.3.1. Definition – extending the discussion of analyticity of the Fourier transform with an exponentially decaying kernel rather than an oscillatory kernel, derivation of inverse as an integral in the complex plane 5.3.2. Convolution Theorem – derivation, examples, application to computing certain classes of definite integrals 5.3.3. Inversion Via Complex Integration 5.3.3.1. Solutions to Ordinary Differential Equations and Rational Transforms – initial conditions, homogeneous and inhomogeneous equations, inversion via the residue theorem 5.3.3.2. Solutions to Partial Differential Equations and Multivalued Transforms – heat equation produces multivalued transforms, evaluation of inverse Laplace transforms to derive solutions 5.4. The Mellin Transform 5.4.1. Definition discussion of strip of convergence, inverse Mellin transform 5.4.2. Convolution Theorem – derivation; NB this will be used in the next chapter 5.4.3. Scaling – expression of scaled integrals in terms of residues 5.5. Problems 6. Asymptotic Analysis 6.1. Definitions 6.1.1. Big-O, Little-O, and The Squiggle – i.e., definitions of asymptotic equivalence in specific limits 6.1.2. Asymptotic Series – definition, properties, numerical calculations, summation acceleration techniques 6.2. Integration by Parts – development of asymptotic series; limitations 6.2.1. Euler-Maclurin Formula – derivation of asymptotic series using integration by parts, application to evaluation of sums 6.3. Watson’s Lemma and h-Transforms – asymptotic behavior of monotonic integrands, application of Mellin transforms in derivation 6.3.1. Application to Complex Integration – evaluation of integrals with branch points at infinity using h-transforms 6.4. Laplace’s Method – asymptotic behavior of nonmonotomic, nonoscillatory integrals 6.5. The Method of Steepest Descents – deriving asymptotic behavior of complex integrals, derive behavior of real integrals by using Cauchy’s Theorem 6.6. Problems

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Book SynopsisComplex plane.- Functions.- Derivative.- Integral.- Analyticity.- Uni cation of holomorphy cauchy theorem and analyticity.- Global cauchy theorem.- Meromorphic functions.- Harmonic functions.- Conformal regions.- Analytic continuation and riemann surfaces.- Uniformization of riemann surfaces.- Complex dynamics.

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Book Synopsis1. Introduction.- 2. Riemann Surfaces, Braids, Mapping Classes, and Teichmueller Theory.- 3. The entropy of surface homeomorphisms.- 4. Conformal invariants of homotopy classes of curves. The Main theorem.- 5. Reducible pure braids. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem.- 6. The general case. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem.- 7. The conformal module and holomorphic families of polynomials.-  8. Gromov's Oka Principle and conformal module.- 9. Gromov's Oka Principle for (g, m)-fiber bundles.- 10. Fundamental groups and bounds for the extremal length.- 11. Counting functions.- 12. Riemann surfaces of second kind and finiteness theorems.- A. Several complex variables.- B. A Lemma on Conjugation.- C. Koebe's Theorem.- Index.- References.

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