Hemant Kumar Pathak

Graduate Texts in Mathem, atics 5Hemant Kumar PathakAn Introduction to Complex Analysis This book presents all useful topics in complex analysis. It contains complete definitions of topics well-explained by suitable examples, written in a lucid manner, with full explanations, and proofs throughout the book. The initial chapters cover the topics such as analytic functions of one complex variable, power series, uniform convergence of sequences and series, complex integration, singularities and principle of arguments, calculus of residuals, bilinear transformations, conformal mappings usually taught at the undergraduate level. The later chapters contain some additional topics such as spaces of analytic functions, entire and meromorphic functions, analytic continuation, harmonic functions, canonical products, the range of an analytic function, univalent functions, analytic functions of several complex variables which are taught at the beginning graduate level. The book has evolved out of teaching over a period of three decades and can serve as a textbook on Complex Analysis. This book is designed as a textbook for a two-semester course in complex analysis for upper undergraduate and beginning graduate students. The book is also accessible to junior mathematics majors who have studied set theory, elementary calculus, and number systems. The benefit most from the book, students should have some prior knowledge of complex numbers. However, the essential pre-requisites are quite minimal, and include basic calculus with some knowledge of partial derivatives, definite integrals, and topics in advanced calculus such as Leibnitz rule for differentiating under the integral sign and to some extent analysis of infinite series. The essential feature of the book lies on the fact that it lays the groundwork for further study in analysis, linear algebra, numerical analysis, differential geometry, physics (including hydromechanics and thermodynamics), and electric engineering. This book is an asset for undergraduate and graduate students of mathematics and engineering.

Graduate Texts in Mathemkatics 2Hemant Kumar PathakGeneral Topology and Applications This textbook discusses all useful topics in general topology and applications. It contains complete definitions of topics well-explained by suitable examples, explanations, and proofs throughout the book. The book studies major topics on topological spaces and continuous functions, countability, separable spaces and connectedness, compactness and the Tychonoff theorem, separation axioms, compactification, products and coproducts, embedding, metrization theorem and paracompactness, nets and filters, complete metric spaces and function spaces, uniform spaces, manifolds and their geometry, Baire spaces and dimension geometry. This book is designed as a textbook for a two-semester course in point-set topology or metric topology, as well as for a first-semester course in topology at the lower undergraduate level. The book is also accessible to junior mathematics majors who have studied multivariable.

Graduate Texts in Mathematics 1Hemant Kumar PathakFunctional Analysis and Applications This textbook discusses all useful topics in functional analysis and applications. It contains complete definitions of topics well-explained by suitable examples, explanations, and proofs throughout the book. The book studies major topics on normed linear spaces, bounded linear functionals, Banach spaces, Hahn-Banach theorem, reflexive spaces, open mapping theorem, closed graph theorem, Banach Steinhaus theorem, spaces of bounded linear functionals, weak and weak* topologies, Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma, Banach algebra, the spectrum and the Gelfand-Mazur theorem, commutative Banach algebra, differentiation and integration In Banach spaces, variational methods and optimizations, approximation and optimization, operator theory, nonlinear operators and applications, spectral theory, Sturm-Liouville system and Fredholm alternative, fixed point theory, Banach contraction principle, Brouwer's fixed point theorem, Schauder's fixed point theorem, variational inequalities and applications, frames and bases in Hilbert spaces, frames and bases in Banach spaces. This book Is designed as a textbook for a two-semester course in functional analysis, as well as for a first semester course in functional analysis at the lower undergraduate level. The book Is also accessible to junior mathematics majors who have studied advanced calculus or multivariable calculus. The essential prerequisites for reading this book are quite minimal: not much more than a stiff course in analysis, advanced calculus, and abstract algebra.

This book offers an essential textbook on complex analysis. After introducing the theory of complex analysis, it places special emphasis on the importance of Poincare theorem and Hartog’s theorem in the function theory of several complex variables. Further, it lays the groundwork for future study in analysis, linear algebra, numerical analysis, geometry, number theory, physics (including hydrodynamics and thermodynamics), and electrical engineering. To benefit most from the book, students should have some prior knowledge of complex numbers. However, the essential prerequisites are quite minimal, and include basic calculus with some knowledge of partial derivatives, definite integrals, and topics in advanced calculus such as Leibniz’s rule for differentiating under the integral sign and to some extent analysis of infinite series. The book offers a valuable asset for undergraduate and graduate students of mathematics and engineering, as well as students with no background in topological properties.

This book systematically introduces the theory of nonlinear analysis, providing an overview of topics such as geometry of Banach spaces, differential calculus in Banach spaces, monotone operators, and fixed point theorems. It also discusses degree theory, nonlinear matrix equations, control theory, differential and integral equations, and inclusions. The book presents surjectivity theorems, variational inequalities, stochastic game theory and mathematical biology, along with a large number of applications of these theories in various other disciplines. Nonlinear analysis is characterised by its applications in numerous interdisciplinary fields, ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. Organised into ten chapters, the book shows the elegance of the subject and its deep-rooted concepts and techniques, which provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in diverse applied fields. It is intended for graduate and undergraduate students of mathematics and engineering who are familiar with discrete mathematical structures, differential and integral equations, operator theory, measure theory, Banach and Hilbert spaces, locally convex topological vector spaces, and linear functional analysis.

This book systematically introduces the theory of nonlinear analysis, providing an overview of topics such as geometry of Banach spaces, differential calculus in Banach spaces, monotone operators, and fixed point theorems. It also discusses degree theory, nonlinear matrix equations, control theory, differential and integral equations, and inclusions. The book presents surjectivity theorems, variational inequalities, stochastic game theory and mathematical biology, along with a large number of applications of these theories in various other disciplines. Nonlinear analysis is characterised by its applications in numerous interdisciplinary fields, ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. Organised into ten chapters, the book shows the elegance of the subject and its deep-rooted concepts and techniques, which provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in diverse applied fields. It is intended for graduate and undergraduate students of mathematics and engineering who are familiar with discrete mathematical structures, differential and integral equations, operator theory, measure theory, Banach and Hilbert spaces, locally convex topological vector spaces, and linear functional analysis.

Functions of a Complex Variable provides all the material for a course on the theory of functions of a complex variable at the senior undergraduate and beginning graduate level. Also suitable for self-study, the book covers every topic essential to training students in complex analysis. It also incorporates special topics to enhance students’ understanding of the subject, laying the foundation for future studies in analysis, linear algebra, numerical analysis, geometry, number theory, physics, thermodynamics, or electrical engineering.After introducing the basic concepts of complex numbers and their geometrical representation, the text describes analytic functions, power series and elementary functions, the conformal representation of an analytic function, special transformations, and complex integration. It next discusses zeros of an analytic function, classification of singularities, and singularity at the point of infinity; residue theory, principle of argument, Rouché’s theorem, and the location of zeros of complex polynomial equations; and calculus of residues, emphasizing the techniques of definite integrals by contour integration. The authors then explain uniform convergence of sequences and series involving Parseval, Schwarz, and Poisson formulas. They also present harmonic functions and mappings, inverse mappings, and univalent functions as well as analytic continuation.