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BL
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| Record Number
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890511
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| Main Entry
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Pathak, Hemant Kumar
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| Title & Author
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Complex analysis and applications /\ Hemant Kumar Pathak.
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| Publication Statement
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Singapore :: Springer,, 2019.
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| Page. NO
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1 online resource (xxv, 928 pages) :: illustrations
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| ISBN
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9789811397332
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: 9789811397349
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: 9789811397356
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: 9811397333
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: 9811397341
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: 981139735X
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9789811397332
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Contents; About the Author; Acronyms; Glossary of Symbols; 1 Complex Numbers and Metric Topology of mathbbC; 1.1 Introduction; 1.2 Complex Numbers; 1.2.1 Equality of Complex Numbers; 1.2.2 Fundamental Laws of Addition and Multiplication; 1.2.3 Difference and Division of Two Complex Numbers; 1.3 Modulus and Argument of Complex Numbers; 1.4 Geometrical Representations of Complex Numbers; 1.5 Modulus and Argument of Complex Numbers; 1.5.1 Polar Forms of Complex Numbers; 1.5.2 Conjugates; 1.5.3 Vector Representation of Complex Numbers; 1.5.4 Multiplication of a Complex Number by i
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1.14.2 Dense Set1.14.3 Connectedness; 1.14.4 Convergence and Completeness; 1.14.5 Component; 1.14.6 Compactness; 1.14.7 Continuity; 1.14.8 Topological Spaces; 1.14.9 Metrizable Spaces; 1.14.10 Homeomorphism; 2 Analytic Functions, Power Series, and Uniform Convergence; 2.1 Introduction; 2.2 Functions of Complex Variables; 2.2.1 Limits of Functions; 2.2.2 Continuity; 2.3 Uniform Continuity; 2.4 Differentiability; 2.5 Analytic and Regular Functions; 2.6 Cauchy-Riemann Equations; 2.6.1 Conjugate Functions; 2.6.2 Harmonic Functions; 2.6.3 Polar Form of the Cauchy-Riemann Equations
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1.6 Properties of Moduli1.7 Properties of Arguments; 1.8 Equations of Straight Lines; 1.9 Equations of Circles; 1.9.1 General Equation of a Circle; 1.9.2 Equations of Circles Through Three Points; 1.10 Inverse Points; 1.10.1 Inverse Points with Respect to Lines; 1.10.2 Inverse Points with Respect to Circles; 1.11 Relations Between Inverse Points with Respect To Circles; 1.12 Riemann Spheres and Point at Infinity; 1.12.1 Point at Infinity; 1.12.2 Riemann Spheres; 1.13 Cauchy-Schwarz's Inequality and Lagrange's Identity; 1.14 Metric Spaces and Topology of mathbbC; 1.14.1 Metric Spaces
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2.12 The Logarithm and Power Functions2.13 The Riemann Surface for Log z; 2.14 Uniform Convergence of a Sequence; 2.14.1 General Principle of Uniform Convergence of a Sequence; 2.15 Uniform Convergence of a Series; 2.15.1 Principle of Uniform Convergence of a Series; 2.15.2 Sufficient Tests for Uniform Convergence of a Series; 2.15.3 Weierstrass M-Test; 2.16 Hardy's Tests for Uniform Convergence; 2.17 Continuity of the Sum Function of a Series; 3 Complex Integrations; 3.1 Introduction; 3.2 Complex Integrations; 3.2.1 Some Definitions; 3.2.2 Rectifiable Curves; 3.3 Complex Integrals
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2.7 Methods of Constructing Analytic Functions2.7.1 Simple Methods of Constructing Analytic Functions (Without Using Integrals); 2.8 Power Series; 2.8.1 Absolute Convergence of a Power Series; 2.8.2 Some Special Test for Convergence of Series; 2.9 Certain Theorems on Power Series; 2.9.1 Abel's Theorem; 2.9.2 Cauchy-Hadamard's Theorem; 2.9.3 Circle and Radius of Convergence of a Power Series; 2.9.4 Analyticity of the Sum Function of a Power Series; 2.9.5 Abel's Limit Theorem; 2.10 Elementary Functions of a Complex Variable; 2.11 Many-Valued Functions: Branches
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| Abstract
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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.
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| Subject
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Functions of complex variables.
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| Subject
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Mathematical analysis.
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| Subject
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Functions of complex variables.
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| Subject
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Mathematical analysis.
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| Dewey Classification
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515/.9
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| LC Classification
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QA331
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