The theory of analytic functions.
Idioma: Inglés Lenguaje original: Ruso Series International monographs on advanced mathematics and physicsEditor: Delhi : Hindustan Pub. Corpn., 1963Descripción: x, 374 p. : diagrs. ; 24 cmTítulos uniformes: Kratkii kurs teorii analiticheskikh funktsii. Inglés Tema(s): Analytic functionsOtra clasificación: 30-01Preface v INTRODUCTION [1] 1. The scope of the theory of analytic functions [1] 2. Analytic functions of a complex variable [2] 1. COMPLEX NUMBERS AND THEIR GEOMETRICAL REPRESENTATION [5] L Geometrical representation of complex numbers on a plane [5] 2. Operations with complex numbers [7] 3. The limit of a sequence [9] 4. The point at infinity and the stereographic projection [10] 5. Point sets in a plane [13] 2. FUNCTIONS OF A COMPLEX VARIABLE. DERIVATIVE AND ITS GEOMETRICAL AND HYDROMECHANICAL INTERPRETATION [17] 1. Functions of a complex variable [17] 2. Limit of a function at a point [17] 3. Continuity [19] 4. Continuous curve [20] 5. Derivative and differential [24] 6. Rules of differentiation [26] 7. Necessary and sufficient conditions for differentiability at an interior point of a domain [27] 8. Geometrical interpretation of the argument of a derivative [33] 9. Geometrical interpretation of the modulus of a derivative [35] 10. Example: linear and bilinear functions [36] 11. The angle formed at the point at infinity [37] 12. Harmonic and conjugate harmonic functions [39] 13. Hydromechanical interpretation of an analytic function [43] 14. Examples [49] 3. ELEMENTARY ANALYTIC FUNCTIONS AND THE CORRESPONDING CONFORMAL MAPPINGS [52] 1. The polynomial [52] 2. Points at which the conformality of the mapping fails [53] 3. Mapping of the form w = (z—a)n [54] 4. Group properties of the bilinear transformations [58] 5. The circle-preserving property [61] 6. Invariance of the cross ratio [64] 7. Mapping of domains bounded by straight lines or circles [70] 8. Symmetry and its preservation [72] 9. Examples [76] 10. The Joukowski function [79] 11. Definition of an exponential function [84] 12. Mapping by an exponential function [86] 13. Trigonometric functions [91] 14. Geometrical behaviour [96] 15.. Continuation [100] 16. One-valued branches of the many-valued functions [102] 17. The function w = n√z [104] 18. The function w = n√P(z) [109] 19. Logarithm [114] 20. The general power function and exponential function [119] 21. Inverse trigonometric functions [125] 4. COMPLEX TERM SERIES. POWER SERIES [130] 1. Convergent and divergent series [130] 2. The Cauchy-Hadamard theorem [132] 3. Analyticity of the sum of a power series [135] 4. Uniform convergence [138] 5. INTEGRATION OF A FUNCTION OF A COMPLEX VARIABLE [140] 1. Integral of a function of a complex variable [140] 2. The properties of integrals [143] 3. Reduction to the calculation of an ordinary integral [144] 4. The integral theorem of Cauchy [146] 5. The integral theorem of Cauchy : proof (continued) [150] 6. Application to the calculation of definite integrals [152] 7. Integral and the primitive [162] 8. Generalization of the integral theorem of Cauchy when the function is not analytic on the contour of integration [165] 9. The theorem on a composite contour [166] 10. Integral as a function of a point in a multiply-connected domain [169] 6. THE CAUCHY INTEGRAL FORMULA AND ITS CONSEQUENCES [173] 1. The Cauchy integral formula [173] 2. Decomposition of an analytic function into a power series. The theorem of Liouville [175] 3. Infinite differentiability of analytic and harmonic functions [178] 4. The theorem of Morera [182] 5. Weierstrass’s theorem on the uniformly convergent series of analytic functions [183] 6. The uniqueness theorem [187] 7. A -points and, in particular, the zeros [190] 8. Series of power series [191] 9. Substitution of one series in another [194] 10. Division of the power series [198] 11. Expansion of the functions cot z, tan z, cosec z and sec z into power series [204] 12. Expansion of harmonic functions into series. The Poisson integral and the Schwartz formula [208] 7. THE LAURENT SERIES. ISOLATED SINGULAR POINTS OF A SIMPLE CHARACTER. ENTIRE AND MEROMORPHIC FUNCTIONS [215] 1. The Laurent series [215] 2. Laurent’s theorem [219] 3. Isolated singular points of a simple character [222] 4. The theorem of Sokhotskii [227] 5. Singularities of the derivatives and of the rational combinations of analytic functions [231] 6. The case of the point at infinity [235] 7. Entire and meromorphic functions [236] 8. Expansion of an entire function into a product [241] 9. The order and the type of an entire function [248] 8. RESIDUES AND THEIR APPLICATIONS. THE PRINCIPLE OF THE ARGUMENT [251] 1. The theorem of residues and its application to the evaluation of definite integral [251] 2. The principle of the argument and its consequences [258] 3. Residue at the point at infinity [265] 4. Application of the theorem of residues to the expansion of meromorphic functions into partial fractions [267] 5. Expansion of sec z, cot z, cosec z and tan z into partial fractions [273] 9. ANALYTIC CONTINUATION. THE CONCEPT OF A RIEMANN SURFACE. SINGULARITIES [283] 1. Analytic continuation [283] 2. Direct analytic continuation [285] 3. The construction of an analytic function from its elements [287] 4. Construction of a Riemann surface [288] 5. The Riemann-Schwartz principle of symmetry [291] 6. Singularities on the boundary of the circle of convergence of a power series [295] 7. Criterion for locating singularities [300] 8. Determination of the radius of convergence of a power series from a known distribution of singularities of the function [304] 9. Isolated singularities of a multiple character [309] 10. MAPPINGS BY ANALYTIC FUNCTIONS. THE CONCEPT OF ELLIPTIC FUNCTIONS. THE CHRISTOFFEL-SCHWARTZ FORMULA [315] 1. Mapping of a domain by an analytic function [315] 2. The principle of the maximum modulus and the Schwartz lemma [316] 3. Local criterion of single-sheetedness [318] 4. Inversion of an analytic function [320] 5. The extension of the concept of single-sheetedness to functions having poles [324] 6. The concept of the Riemann theorem. The uniqueness of mapping [326] 7. The concept of the correspondence between boundaries. The converse theorem [327] 8. Mapping of the upper half-plane by the elliptic integral [334] 9. The concept of the Jacobian elliptic function sn w [339] 10. The Christoffel-Schwartz integral [344] 11. Flow past a circular cylinder (without circulation) [351] 12. Hydromechanical interpretation of the simplest singularities [353] 13. The general solution of the problem of flow past a circular cylinder [358] 14. Determination of the lift of an aerofoil [363] References [369] Index [372]
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"Translated from the original Russian title 'Kratkii kurs teorii analiticheskikh funktsii.'"
Bibliografía: p. [369]-371.
Preface v --
INTRODUCTION [1] --
1. The scope of the theory of analytic functions [1] --
2. Analytic functions of a complex variable [2] --
1. COMPLEX NUMBERS AND THEIR GEOMETRICAL REPRESENTATION [5] --
L Geometrical representation of complex numbers on a plane [5] --
2. Operations with complex numbers [7] --
3. The limit of a sequence [9] --
4. The point at infinity and the stereographic projection [10] --
5. Point sets in a plane [13] --
2. FUNCTIONS OF A COMPLEX VARIABLE. DERIVATIVE AND ITS GEOMETRICAL AND HYDROMECHANICAL INTERPRETATION [17] --
1. Functions of a complex variable [17] --
2. Limit of a function at a point [17] --
3. Continuity [19] --
4. Continuous curve [20] --
5. Derivative and differential [24] --
6. Rules of differentiation [26] --
7. Necessary and sufficient conditions for differentiability at an interior point of a domain [27] --
8. Geometrical interpretation of the argument of a derivative [33] --
9. Geometrical interpretation of the modulus of a derivative [35] --
10. Example: linear and bilinear functions [36] --
11. The angle formed at the point at infinity [37] --
12. Harmonic and conjugate harmonic functions [39] --
13. Hydromechanical interpretation of an analytic function [43] --
14. Examples [49] --
3. ELEMENTARY ANALYTIC FUNCTIONS AND THE CORRESPONDING CONFORMAL MAPPINGS [52] --
1. The polynomial [52] --
2. Points at which the conformality of the mapping fails [53] --
3. Mapping of the form w = (z—a)n [54] --
4. Group properties of the bilinear transformations [58] --
5. The circle-preserving property [61] --
6. Invariance of the cross ratio [64] --
7. Mapping of domains bounded by straight lines or circles [70] --
8. Symmetry and its preservation [72] --
9. Examples [76] --
10. The Joukowski function [79] --
11. Definition of an exponential function [84] --
12. Mapping by an exponential function [86] --
13. Trigonometric functions [91] --
14. Geometrical behaviour [96] --
15.. Continuation [100] --
16. One-valued branches of the many-valued functions [102] --
17. The function w = n√z [104] --
18. The function w = n√P(z) [109] --
19. Logarithm [114] --
20. The general power function and exponential function [119] --
21. Inverse trigonometric functions [125] --
4. COMPLEX TERM SERIES. POWER SERIES [130] --
1. Convergent and divergent series [130] --
2. The Cauchy-Hadamard theorem [132] --
3. Analyticity of the sum of a power series [135] --
4. Uniform convergence [138] --
5. INTEGRATION OF A FUNCTION OF A COMPLEX VARIABLE [140] --
1. Integral of a function of a complex variable [140] --
2. The properties of integrals [143] --
3. Reduction to the calculation of an ordinary integral [144] --
4. The integral theorem of Cauchy [146] --
5. The integral theorem of Cauchy : proof (continued) [150] --
6. Application to the calculation of definite integrals [152] --
7. Integral and the primitive [162] --
8. Generalization of the integral theorem of Cauchy when the function is not analytic on the contour of integration [165] --
9. The theorem on a composite contour [166] --
10. Integral as a function of a point in a multiply-connected domain [169] --
6. THE CAUCHY INTEGRAL FORMULA AND ITS CONSEQUENCES [173] --
1. The Cauchy integral formula [173] --
2. Decomposition of an analytic function into a power series. The theorem of Liouville [175] --
3. Infinite differentiability of analytic and harmonic functions [178] --
4. The theorem of Morera [182] --
5. Weierstrass’s theorem on the uniformly convergent series of analytic functions [183] --
6. The uniqueness theorem [187] --
7. A -points and, in particular, the zeros [190] --
8. Series of power series [191] --
9. Substitution of one series in another [194] --
10. Division of the power series [198] --
11. Expansion of the functions cot z, tan z, cosec z and sec z into power series [204] --
12. Expansion of harmonic functions into series. The Poisson integral and the Schwartz formula [208] --
7. THE LAURENT SERIES. ISOLATED SINGULAR POINTS OF A SIMPLE CHARACTER. ENTIRE AND MEROMORPHIC FUNCTIONS [215] --
1. The Laurent series [215] --
2. Laurent’s theorem [219] --
3. Isolated singular points of a simple character [222] --
4. The theorem of Sokhotskii [227] --
5. Singularities of the derivatives and of the rational combinations of analytic functions [231] --
6. The case of the point at infinity [235] --
7. Entire and meromorphic functions [236] --
8. Expansion of an entire function into a product [241] --
9. The order and the type of an entire function [248] --
8. RESIDUES AND THEIR APPLICATIONS. THE PRINCIPLE OF THE ARGUMENT [251] --
1. The theorem of residues and its application to the evaluation of definite integral [251] --
2. The principle of the argument and its consequences [258] --
3. Residue at the point at infinity [265] --
4. Application of the theorem of residues to the expansion of meromorphic functions into partial fractions [267] --
5. Expansion of sec z, cot z, cosec z and tan z into partial fractions [273] --
9. ANALYTIC CONTINUATION. THE CONCEPT OF A RIEMANN SURFACE. SINGULARITIES [283] --
1. Analytic continuation [283] --
2. Direct analytic continuation [285] --
3. The construction of an analytic function from its elements [287] --
4. Construction of a Riemann surface [288] --
5. The Riemann-Schwartz principle of symmetry [291] --
6. Singularities on the boundary of the circle of convergence of a power series [295] --
7. Criterion for locating singularities [300] --
8. Determination of the radius of convergence of a power series from a known distribution of singularities of the function [304] --
9. Isolated singularities of a multiple character [309] --
10. MAPPINGS BY ANALYTIC FUNCTIONS. THE CONCEPT OF ELLIPTIC FUNCTIONS. THE CHRISTOFFEL-SCHWARTZ FORMULA [315] --
1. Mapping of a domain by an analytic function [315] --
2. The principle of the maximum modulus and the Schwartz lemma [316] --
3. Local criterion of single-sheetedness [318] --
4. Inversion of an analytic function [320] --
5. The extension of the concept of single-sheetedness to functions having poles [324] --
6. The concept of the Riemann theorem. The uniqueness of mapping [326] --
7. The concept of the correspondence between boundaries. The converse theorem [327] --
8. Mapping of the upper half-plane by the elliptic integral [334] --
9. The concept of the Jacobian elliptic function sn w [339] --
10. The Christoffel-Schwartz integral [344] --
11. Flow past a circular cylinder (without circulation) [351] --
12. Hydromechanical interpretation of the simplest singularities [353] --
13. The general solution of the problem of flow past a circular cylinder [358] --
14. Determination of the lift of an aerofoil [363] --
References [369] --
Index [372] --
MR, 35 #6798
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