TY  - BOOK
AU  - Lang,Serge
TI  - Complex analysis
T2  - Graduate texts in mathematics
SN  - 9783540780595
AV  - QA331.7 .L36 1993
U1  - 515/.9 20
PY  - 1993///
CY  - New York
PB  - Springer-Verlag
KW  - Functions of complex variables
KW  - Mathematical analysis
N1  - Includes bibliographical references (p. [454]) and index; Pt. 1; Basic Theory; Ch. I; Complex Numbers and Functions; 1; Definition; 2; Polar Form; 3; Complex Valued Functions; 4; Limits and Compact Sets; Compact Sets; 5; Complex Differentiability; 6; The Cauchy-Riemann Equations; 7; Angles Under Holomorphic Maps; Ch. II; Power Series; 1; Formal Power Series; 2; Convergent Power Series; 3; Relations Between Formal and Convergent Series; Sums and Products; Quotients; Composition of Series; 4; Analytic Functions; 5; Differentiation of Power Series; 6; The Inverse and Open Mapping Theorems; 7; The Local Maximum Modulus Principle; Ch. III; Cauchy's Theorem, First Part; 1; Holomorphic Functions on Connected Sets; Appendix: Connectedness; 2; Integrals Over Paths; 3; Local Primitive for a Holomorphic Function; 4; Another Description of the Integral Along a Path; 5; The Homotopy Form of Cauchy's Theorem; 6; Existence of Global Primitives. Definition of the Logarithm; 7; The Local Cauchy Formula; Ch. IV; Winding Numbers and Cauchy's Theorem; 1; The Winding Number; 2; The Global Cauchy Theorem; Dixon's Proof of Theorem 2.5 (Cauchy's Formula); 3; Artin's Proof; Ch. V; Applications of Cauchy's Integral Formula; 1; Uniform Limits of Analytic Functions; 2; Laurent Series; 3; Isolated Singularities; Removable Singularities; Poles; Essential Singularities; Ch. VI; Calculus of Residues; 1; The Residue Formula; Residues of Differentials; 2; Evaluation of Definite Integrals; Fourier Transforms; Trigonometric Integrals; Mellin Transforms; Ch. VII; Conformal Mappings; 1; Schwarz Lemma; 2; Analytic Automorphisms of the Disc; 3; The Upper Half Plane; 4; Other Examples; 5; Fractional Linear Transformations; Ch. VIII; Harmonic Functions; 1; Definition; Application: Perpendicularity; Application: Flow Lines; 2; Examples; 3; Basic Properties of Harmonic Functions; 4; The Poisson Formula; 5; Construction of Harmonic Functions --; Pt. 2; Geometric Function Theory; Ch. IX; Schwarz Reflection; 1; Schwarz Reflection (by Complex Conjugation); 2; Reflection Across Analytic Arcs; 3; Application of Schwarz Reflection; Ch. X; The Riemann Mapping Theorem; 1; Statement of the Theorem; 2; Compact Sets in Function Spaces; 3; Proof of the Riemann Mapping Theorem; 4; Behavior at the Boundary; Ch. XI; Analytic Continuation Along Curves; 1; Continuation Along a Curve; 2; The Dilogarithm; 3; Application to Picard's Theorem --; Pt. 3; Various Analytic Topics; Ch. XII; Applications of the Maximum Modulus Principle and Jensen's Formula; 1; Jensen's Formula; 2; The Picard-Borel Theorem; 3; Bounds by the Real Part, Borel-Caratheodory Theorem; 4; The Use of Three Circles and the Effect of Small Derivatives; Hermite Interpolation Formula; 5; Entire Functions with Rational Values; 6; The Phragmen-Lindelof and Hadamard Theorems; Ch. XIII; Entire and Meromorphic Functions; 1; Infinite Products; 2; Weierstrass Products; 3; Functions of Finite Order; 4; Meromorphic Functions, Mittag-Leffler Theorem; Ch. XIV; Elliptic Functions; 1; The Liouville Theorems; 2; The Weierstrass Function; 3; The Addition Theorem; 4; The Sigma and Zeta Functions; Ch. XV; The Gamma and Zeta Functions; 1; The Differentiation Lemma; 2; The Gamma Function; Weierstrass Product; The Mellin Transform; Proof of Stirling's Formula; 3; The Lerch Formula; 4; Zeta Functions; Ch. XVI; The Prime Number Theorem; 1; Basic Analytic Properties of the Zeta Function; 2; The Main Lemma and its Application; 3; Proof of the Main Lemma; App. 1. Summation by Parts and Non-Absolute Convergence --; App. 2. Difference Equations --; App. 3. Analytic Differential Equations --; App. 4. Fixed Points of a Fractional Linear Transformation --; App. 5. Cauchy's Formula for C[actual symbol not reproducible] Functions
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