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1.

図書

図書
Joseph Bak, Donald J. Newman
出版情報: New York : Springer-Verlag, c1997  x, 294 p. ; 25 cm
シリーズ名: Undergraduate texts in mathematics
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2.

電子ブック

EB
Joseph Bak, Donald J. Newman
出版情報: SpringerLink Books Complete , Springer New York, 2010
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目次情報: 続きを見る
Preface to the Third Edition
Preface to the Second Edition
The Complex Numbers / 1:
Introduction
The Field of Complex Numbers / 1.1:
The Complex Plane / 1.2:
The Solution of the Cubic Equation / 1.3:
Topological Aspects of the Complex Plane / 1.4:
Stereographic Projection; The Point at Infinity / 1.5:
Exercises
Functions of the Complex Variable z / 2:
Analytic Polynomials / 2.1:
Power Series / 2.2:
Differentiability and Uniqueness of Power Series / 2.3:
Analytic Functions / 3:
Analyticity and the Cauchy-Riemann Equations / 3.1:
Line Integrals and Entire Functions / 3.2:
Properties of the Line Integral / 4.1:
The Closed Curve Theorem for Entire Functions / 4.2:
Properties of Entire Functions / 5:
The Cauchy Integral Formula and Taylor Expansion for Entire Functions / 5.1:
Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem / 5.2:
Newton's Method and Its Application to Polynomial Equations / 5.3:
Properties of Analytic Functions / 6:
The Power Series Representation for Functions Analytic in a Disc / 6.1:
Analytic in an Arbitrary Open Set / 6.2:
The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points / 6.3:
Further Properties of Analytic Functions / 7:
The Open Mapping Theorem; Schwarz' Lemma / 7.1:
The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reflection Principle and Analytic Arcs / 7.2:
Simply Connected Domains / 8:
The General Cauchy Closed Curve Theorem / 8.1:
The Analytic Function log z / 8.2:
Isolated Singularities of an Analytic Function / 9:
Classification of Isolated Singularities; Riemann's Principle and the Casorati-Weierstrass Theorem / 9.1:
Laurent Expansions / 9.2:
The Residue Theorem / 10:
Winding Numbers and the Cauchy Residue Theorem / 10.1:
Applications of the Residue Theorem / 10.2:
Applications of the Residue Theorem to the Evaluation of Integrals and Sums / 11:
Evaluation of Definite Integrals by Contour Integral Techniques / 11.1:
Application of Contour Integral Methods to Evaluation and Estimation of Sums / 11.2:
Further Contour Integral Techniques / 12:
Shifting the Contour of Integration / 12.1:
An Entire Function Bounded in Every Direction / 12.2:
Introduction to Conformal Mapping / 13:
Conformal Equivalence / 13.1:
Special Mappings / 13.2:
Schwarz-Christoffel Transformations / 13.3:
The Riemann Mapping Theorem / 14:
Conformal Mapping and Hydrodynamics / 14.1:
Mapping Properties of Analytic Functions on Closed Domains / 14.2:
Maximum-Modulus Theorems for Unbounded Domains / 15:
A General Maximum-Modulus Theorem / 15.1:
The Phragmén-Lindelöf Theorem / 15.2:
Harmonic Functions / 16:
Poisson Formulae and the Dirichlet Problem / 16.1:
Liouville Theorems for Re f; Zeroes of Entire Functions of Finite Order / 16.2:
Different Forms of Analytic Functions / 17:
Infinite Products / 17.1:
Analytic Functions Defined by Definite Integrals / 17.2:
Analytic Functions Defined by Dirichlet Series / 17.3:
Analytic Continuation; The Gamma and Zeta Functions / 18:
Analytic Continuation of Dirichlet Series / 18.1:
The Gamma and Zeta Functions / 18.3:
Applications to Other Areas of Mathematics / 19:
A Variation Problem / 19.1:
The Fourier Uniqueness Theorem / 19.2:
An Infinite System of Equations / 19.3:
Applications to Number Theory / 19.4:
An Analytic Proof of The Prime Number Theorem / 19.5:
Answers
References
Appendices
Index
Preface to the Third Edition
Preface to the Second Edition
The Complex Numbers / 1:
3.

図書

図書
Joseph Bak, Donald J. Newman
出版情報: New York : Springer, c2010  xii, 328 p. ; 25 cm
シリーズ名: Undergraduate texts in mathematics
所蔵情報: loading…
目次情報: 続きを見る
Preface to the Third Edition
Preface to the Second Edition
The Complex Numbers / 1:
Introduction
The Field of Complex Numbers / 1.1:
The Complex Plane / 1.2:
The Solution of the Cubic Equation / 1.3:
Topological Aspects of the Complex Plane / 1.4:
Stereographic Projection; The Point at Infinity / 1.5:
Exercises
Functions of the Complex Variable z / 2:
Analytic Polynomials / 2.1:
Power Series / 2.2:
Differentiability and Uniqueness of Power Series / 2.3:
Analytic Functions / 3:
Analyticity and the Cauchy-Riemann Equations / 3.1:
Line Integrals and Entire Functions / 3.2:
Properties of the Line Integral / 4.1:
The Closed Curve Theorem for Entire Functions / 4.2:
Properties of Entire Functions / 5:
The Cauchy Integral Formula and Taylor Expansion for Entire Functions / 5.1:
Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem / 5.2:
Newton's Method and Its Application to Polynomial Equations / 5.3:
Properties of Analytic Functions / 6:
The Power Series Representation for Functions Analytic in a Disc / 6.1:
Analytic in an Arbitrary Open Set / 6.2:
The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points / 6.3:
Further Properties of Analytic Functions / 7:
The Open Mapping Theorem; Schwarz' Lemma / 7.1:
The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reflection Principle and Analytic Arcs / 7.2:
Simply Connected Domains / 8:
The General Cauchy Closed Curve Theorem / 8.1:
The Analytic Function log z / 8.2:
Isolated Singularities of an Analytic Function / 9:
Classification of Isolated Singularities; Riemann's Principle and the Casorati-Weierstrass Theorem / 9.1:
Laurent Expansions / 9.2:
The Residue Theorem / 10:
Winding Numbers and the Cauchy Residue Theorem / 10.1:
Applications of the Residue Theorem / 10.2:
Applications of the Residue Theorem to the Evaluation of Integrals and Sums / 11:
Evaluation of Definite Integrals by Contour Integral Techniques / 11.1:
Application of Contour Integral Methods to Evaluation and Estimation of Sums / 11.2:
Further Contour Integral Techniques / 12:
Shifting the Contour of Integration / 12.1:
An Entire Function Bounded in Every Direction / 12.2:
Introduction to Conformal Mapping / 13:
Conformal Equivalence / 13.1:
Special Mappings / 13.2:
Schwarz-Christoffel Transformations / 13.3:
The Riemann Mapping Theorem / 14:
Conformal Mapping and Hydrodynamics / 14.1:
Mapping Properties of Analytic Functions on Closed Domains / 14.2:
Maximum-Modulus Theorems for Unbounded Domains / 15:
A General Maximum-Modulus Theorem / 15.1:
The Phragmén-Lindelöf Theorem / 15.2:
Harmonic Functions / 16:
Poisson Formulae and the Dirichlet Problem / 16.1:
Liouville Theorems for Re f; Zeroes of Entire Functions of Finite Order / 16.2:
Different Forms of Analytic Functions / 17:
Infinite Products / 17.1:
Analytic Functions Defined by Definite Integrals / 17.2:
Analytic Functions Defined by Dirichlet Series / 17.3:
Analytic Continuation; The Gamma and Zeta Functions / 18:
Analytic Continuation of Dirichlet Series / 18.1:
The Gamma and Zeta Functions / 18.3:
Applications to Other Areas of Mathematics / 19:
A Variation Problem / 19.1:
The Fourier Uniqueness Theorem / 19.2:
An Infinite System of Equations / 19.3:
Applications to Number Theory / 19.4:
An Analytic Proof of The Prime Number Theorem / 19.5:
Answers
References
Appendices
Index
Preface to the Third Edition
Preface to the Second Edition
The Complex Numbers / 1:
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