Functions of a complex variable / Gino Moretti.
Series Prentice-Hall international series in applied mathematicsEditor: Englewood Cliffs, N.J. : Prentice-Hall, c1964Descripción: xv, 456 p. : il. ; 24 cmOtra clasificación: 30-01CONTENTS I INTRODUCTION 1.1 FUNCTIONS, [1] 1.11 Functions Defined by Series Expansions, [2] 1.12 Functions Defined by Integrals, [4] 1.2 CONTINUITY, [5] 1.21 Discontinuities, [6] 1.22 Open and Closed Intervals, [9] 1.3 UNIFORMITY, [9] 1.31 Uniform Convergence of a Function to a Limit, [11] 1.32 Uniform Convergence of a Series of Functions, [12] 1.33 An Example of a Nonuniformly Convergent Series, [13] 1.34 Uniform Convergence and Continuity, [13] 1.4 IMPROPER AND INFINITE INTEGRALS, [15] 1.41 Absolute and Uniform Convergence of Improper and Infinite Integrals, [18] 1.42 Weierstrass’ M-test, [19] 1.5 PROBLEMS, [20] 2 COMPLEX NUMBERS [26] 2.1 DEFINITIONS, [26] 2.11 Equality, [27] 2.12 Conjugate, [28] 2.13 Addition, [28] 2.14 Product, [28] 2.15 Quotient, [29] 2.16 Powers with Integral Exponents, [30] 2.17 Roots with Integral Index, [31] 2.18 Concluding Remarks, [32] 2.2 THE COMPLEX PLANE, [32] 2.21 Domains in the Complex Plane, [35] 2.3 PROBLEMS, [36] FUNCTIONS OF A COMPLEX VARIABLE [39] 3.1 INTRODUCTION, [39] 3.11 Plotting, [39] 3.12 Modular Surface, [42] 3.13 Maxwell’s Graphical Method, [43] 3.2 LIMITS, [44] 3.21 Continuity, [45] 3.3 DIFFERENTIATION, [46] 3.31 Analytic Functions and Harmonic Functions, [49] 3.32 A Geometrical Property, [50] 3.33 Holomorphic Functions, [51] 3.4 PROBLEMS, [51] 3.5 COMPLEX INTEGRATION, [54] 3.51 Cauchy’s Theorem, [56] 3.52 Outline of a Proof of Cauchy’s Theorem, [57] 3.53 Integrals of Functions Holomorphic in Non-simply Connected Domains, [58] 3.54 The Integral as the Inverse of the Derivative, [62] 3.55 Cauchy’s Integral, [63] 3.6 A PROPERTY OF THE MODULUS OF A HOLOMORPHIC FUNCTION, [65] 3.61 A Property of Uniqueness, [66] 3.7 DERIVATIVE OF A HOLOMORPHIC FUNCTION, [67] 3.8 PROBLEMS, [68] 3.9 EXAMPLES OF PHYSICAL PROBLEMS WHICH CAN BE DESCRIBED IN TERMS OF ANALYTIC FUNCTIONS, [71] 3.91 Two-dimensional Hydrodynamics, [71] 3.92 Electrostatic Field, [73] 4 ELEMENTARY TRANSCENDENTAL FUNCTIONS [75] 4.1 INTRODUCTION, [75] 4.2 EXPONENTIAL FUNCTION, [75] 4.3 LOGARITHM, [77] 4.4 TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS, [79] 4.5 INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS, [81] 4.6 PROBLEMS, [85] 5 POWER SERIES [87] 5.1 INTRODUCTION, [87] 5.2 SERIES OF FUNCTIONS, [89] 5.3 POWER SERIES, [89] 5.31 Circle of Convergence, [90] 5.32 Behavior of the Series on the Circle of Convergence, [91] 5.4 INTEGRATION OF POWER SERIES, [92] 5.5 DIFFERENTIATION OF POWER SERIES, [93] 5.6 POWER SERIES OF HOLOMORPHIC FUNCTIONS, [94] 5.61 Taylor Series: Uniqueness of the Expansion, [95] 5.62 Analytic Continuation, [97] 5.7 LAURENT SERIES, [99] 5.8 A TABLE OF POWER SERIES EXPANSIONS OF FUNDAMENTAL FUNCTIONS, [100] 5.9 PROBLEMS, [101] SINGULAR POINTS [108] 6.1 ORDINARY AND SINGULAR POINTS, [108] 6.11 Isolated Singularities, [108] 6.12 Behavior at Infinity, [108] 6.2 CLASSIFICATION OF SINGULAR POINTS, [109] 6.21 Poles, [109] 6.22 Essential Singularities, [110] 6.3 RESIDUES, [111] 6.31 Theorem of Residues, [111] 6.32 Residues at Infinity, [111] 6.33 Evaluation of Residues, [112] 6.4 BRANCH POINTS: MANY-VALUED FUNCTIONS, [113] 6.41 Cute, [117] 6.42 Riemann’s Surface, [118] 6.5 PROBLEMS, [119] 6.6 THE ROLE OF SINGULARITIES IN PHYSICAL PROBLEMS, [130] 6.61 Volume Flow and Circulation, [131] 6.62 Sources, Sinks, and Vortices, [131] 6.63 Electrostatic Charges and Magnetic Fields Induced by Currents, [133] 6.64 Higher Order Singularities, [133] 6.65 Hydrodynamic Meaning of Cauchy’s Integral, [134] 6.66 Problems, [134] 6.7 STUDY OF SINGLE-VALUED ANALYTIC FUNCTIONS FROM THE STANDPOINT OF THEIR SINGULARITIES, [138] 6.71 Meromorphic Functions, [138] 6.72 Liouville’s Theorem, [139] 6.73 A Condition Sufficient for the Identity of Two Holomorphic Functions, [139] 6.74 Rational Functions, [139] 6.75 Classification of the Single-valued Functions, [140] 6.76 General Form for Functions with a Finite Number of Singularities, [141] 6.77 Construction of Meromorphic Functions, [141] 6.78 Problems, [143] 7 APPLICATIONS OF THE COMPLEX INTEGRALS [145] 7.1 EVALUATION OF DEFINITE INTEGRALS OF FUNCTIONS OF A REAL VARIABLE, [145] 7.11 Integrals of Continuous Functions between — ∞ and 4-∞, [145] 7.12 Other Integrals between Infinite Limits, [146] 7.13 Integrals Containing Sines and Cosines, [147] 7.14 Integrals of Rational Functions of Sines and Cosines between — π and π, [148] 7.15 Integrals of Real Functions with Simple Poles, [148] 7.16 Definite Integrals of Many-valued Real Functions, [150] 7.2 ASYMPTOTIC EXPANSIONS, [151] 7.21 Properties of Asymptotic Expansions, [155] 7.22 Method of the Steepest Descent, [155] 7.3 A METHOD TO FIND THE SUM OF CERTAIN SERIES, [159] 7.4 ROOTS OF A POLYNOMIAL: THE FUNDAMENTAL THEOREM OF ALGEBRA, [160] 7.41 Number of the Roots of a Polynomial with a Positive Real Part, [161] 7.5 PROBLEMS, [162]
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CONTENTS --
I INTRODUCTION --
1.1 FUNCTIONS, [1] --
1.11 Functions Defined by Series Expansions, [2] --
1.12 Functions Defined by Integrals, [4] --
1.2 CONTINUITY, [5] --
1.21 Discontinuities, [6] --
1.22 Open and Closed Intervals, [9] --
1.3 UNIFORMITY, [9] --
1.31 Uniform Convergence of a Function to a Limit, [11] --
1.32 Uniform Convergence of a Series of Functions, [12] --
1.33 An Example of a Nonuniformly Convergent Series, [13] --
1.34 Uniform Convergence and Continuity, [13] --
1.4 IMPROPER AND INFINITE INTEGRALS, [15] --
1.41 Absolute and Uniform Convergence of Improper and Infinite Integrals, [18] --
1.42 Weierstrass’ M-test, [19] --
1.5 PROBLEMS, [20] --
2 COMPLEX NUMBERS [26] --
2.1 DEFINITIONS, [26] --
2.11 Equality, [27] --
2.12 Conjugate, [28] --
2.13 Addition, [28] --
2.14 Product, [28] --
2.15 Quotient, [29] --
2.16 Powers with Integral Exponents, [30] --
2.17 Roots with Integral Index, [31] --
2.18 Concluding Remarks, [32] --
2.2 THE COMPLEX PLANE, [32] --
2.21 Domains in the Complex Plane, [35] --
2.3 PROBLEMS, [36] --
FUNCTIONS OF A COMPLEX VARIABLE [39] --
3.1 INTRODUCTION, [39] --
3.11 Plotting, [39] --
3.12 Modular Surface, [42] --
3.13 Maxwell’s Graphical Method, [43] --
3.2 LIMITS, [44] --
3.21 Continuity, [45] --
3.3 DIFFERENTIATION, [46] --
3.31 Analytic Functions and Harmonic Functions, [49] --
3.32 A Geometrical Property, [50] --
3.33 Holomorphic Functions, [51] --
3.4 PROBLEMS, [51] --
3.5 COMPLEX INTEGRATION, [54] --
3.51 Cauchy’s Theorem, [56] --
3.52 Outline of a Proof of Cauchy’s Theorem, [57] --
3.53 Integrals of Functions Holomorphic in Non-simply Connected Domains, [58] --
3.54 The Integral as the Inverse of the Derivative, [62] --
3.55 Cauchy’s Integral, [63] --
3.6 A PROPERTY OF THE MODULUS OF A HOLOMORPHIC FUNCTION, [65] --
3.61 A Property of Uniqueness, [66] --
3.7 DERIVATIVE OF A HOLOMORPHIC FUNCTION, [67] --
3.8 PROBLEMS, [68] --
3.9 EXAMPLES OF PHYSICAL PROBLEMS WHICH CAN BE DESCRIBED IN TERMS OF ANALYTIC FUNCTIONS, [71] --
3.91 Two-dimensional Hydrodynamics, [71] --
3.92 Electrostatic Field, [73] --
4 ELEMENTARY TRANSCENDENTAL FUNCTIONS [75] --
4.1 INTRODUCTION, [75] --
4.2 EXPONENTIAL FUNCTION, [75] --
4.3 LOGARITHM, [77] --
4.4 TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS, [79] --
4.5 INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS, [81] --
4.6 PROBLEMS, [85] --
5 POWER SERIES [87] --
5.1 INTRODUCTION, [87] --
5.2 SERIES OF FUNCTIONS, [89] --
5.3 POWER SERIES, [89] --
5.31 Circle of Convergence, [90] --
5.32 Behavior of the Series on the Circle of Convergence, [91] --
5.4 INTEGRATION OF POWER SERIES, [92] --
5.5 DIFFERENTIATION OF POWER SERIES, [93] --
5.6 POWER SERIES OF HOLOMORPHIC FUNCTIONS, [94] --
5.61 Taylor Series: Uniqueness of the Expansion, [95] --
5.62 Analytic Continuation, [97] --
5.7 LAURENT SERIES, [99] --
5.8 A TABLE OF POWER SERIES EXPANSIONS OF FUNDAMENTAL FUNCTIONS, [100] --
5.9 PROBLEMS, [101] --
SINGULAR POINTS [108] --
6.1 ORDINARY AND SINGULAR POINTS, [108] --
6.11 Isolated Singularities, [108] --
6.12 Behavior at Infinity, [108] --
6.2 CLASSIFICATION OF SINGULAR POINTS, [109] --
6.21 Poles, [109] --
6.22 Essential Singularities, [110] --
6.3 RESIDUES, [111] --
6.31 Theorem of Residues, [111] --
6.32 Residues at Infinity, [111] --
6.33 Evaluation of Residues, [112] --
6.4 BRANCH POINTS: MANY-VALUED FUNCTIONS, [113] --
6.41 Cute, [117] --
6.42 Riemann’s Surface, [118] --
6.5 PROBLEMS, [119] --
6.6 THE ROLE OF SINGULARITIES IN PHYSICAL PROBLEMS, [130] --
6.61 Volume Flow and Circulation, [131] --
6.62 Sources, Sinks, and Vortices, [131] --
6.63 Electrostatic Charges and Magnetic Fields Induced by Currents, [133] --
6.64 Higher Order Singularities, [133] --
6.65 Hydrodynamic Meaning of Cauchy’s Integral, [134] --
6.66 Problems, [134] --
6.7 STUDY OF SINGLE-VALUED ANALYTIC FUNCTIONS FROM THE STANDPOINT OF THEIR SINGULARITIES, [138] --
6.71 Meromorphic Functions, [138] --
6.72 Liouville’s Theorem, [139] --
6.73 A Condition Sufficient for the Identity of Two Holomorphic Functions, [139] --
6.74 Rational Functions, [139] --
6.75 Classification of the Single-valued Functions, [140] --
6.76 General Form for Functions with a Finite Number of Singularities, [141] --
6.77 Construction of Meromorphic Functions, [141] --
6.78 Problems, [143] --
7 APPLICATIONS OF THE COMPLEX INTEGRALS [145] --
7.1 EVALUATION OF DEFINITE INTEGRALS OF FUNCTIONS OF A REAL VARIABLE, [145] --
7.11 Integrals of Continuous Functions between — ∞ and 4-∞, [145] --
7.12 Other Integrals between Infinite Limits, [146] --
7.13 Integrals Containing Sines and Cosines, [147] --
7.14 Integrals of Rational Functions of Sines and Cosines between — π and π, [148] --
7.15 Integrals of Real Functions with Simple Poles, [148] --
7.16 Definite Integrals of Many-valued Real Functions, [150] --
7.2 ASYMPTOTIC EXPANSIONS, [151] --
7.21 Properties of Asymptotic Expansions, [155] --
7.22 Method of the Steepest Descent, [155] --
7.3 A METHOD TO FIND THE SUM OF CERTAIN SERIES, [159] --
7.4 ROOTS OF A POLYNOMIAL: THE FUNDAMENTAL THEOREM OF ALGEBRA, [160] --
7.41 Number of the Roots of a Polynomial with a Positive Real Part, [161] --
7.5 PROBLEMS, [162] --
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