Complex Analysis and Applications

Description

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. Front Matter ....Pages i-xxv Complex Numbers and Metric Topology of \(\mathbb {C}\) (Hemant Kumar Pathak)....Pages 1-65 Analytic Functions, Power Series, and Uniform Convergence (Hemant Kumar Pathak)....Pages 67-190 Complex Integrations (Hemant Kumar Pathak)....Pages 191-305 Singularities of Complex Functions and Principle of Argument (Hemant Kumar Pathak)....Pages 307-375 Calculus of Residues and Applications to Contour Integration (Hemant Kumar Pathak)....Pages 377-486 Bilinear Transformations and Applications (Hemant Kumar Pathak)....Pages 487-557 Conformal Mappings and Applications (Hemant Kumar Pathak)....Pages 559-623 Spaces of Analytic Functions (Hemant Kumar Pathak)....Pages 625-648 Entire and Meromorphic Functions (Hemant Kumar Pathak)....Pages 649-713 Analytic Continuation (Hemant Kumar Pathak)....Pages 715-752 Harmonic Functions and Integral Functions (Hemant Kumar Pathak)....Pages 753-805 Canonical Products and Convergence of Entire Functions (Hemant Kumar Pathak)....Pages 807-840 The Range of an Analytic Function (Hemant Kumar Pathak)....Pages 841-858 Univalent Functions and Applications (Hemant Kumar Pathak)....Pages 859-888 Function Theory of Several Complex Variables (Hemant Kumar Pathak)....Pages 889-907 Back Matter ....Pages 909-928