Overview

This famous work is a textbook that emphasizes the conceptual and historical continuity of analytic function theory. The second volume broadens from a textbook to a textbook-treatise, covering the 'canonical' topics (including elliptic functions, entire and meromorphic functions, as well as conformal mapping, etc.) and other topics nearer the expanding frontier of analytic function theory. In the latter category are the chapters on majorization and on functions holomorphic in a half-plane.

Full Product Details

Author:   Einar Hille
Publisher:   American Mathematical Society
Imprint:   Chelsea Publishing
Edition:   2nd Revised edition
Dimensions:   Width: 16.20cm , Height: 2.20cm , Length: 23.00cm
Weight:   0.672kg
ISBN:  

9780828402699

ISBN 10:   0828402698
Pages:   308
Publication Date:   30 January 1959
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Out of Print
Availability:   Awaiting stock  

Table of Contents

1. Number systems: 1.1 The real number system; 1.2 Further properties of real numbers; 1.3 The complex number system 2. The complex plane: 2.1 Geometry of complex numbers; 2.2 Curves and regions in the complex plane; 2.3 Regions and convexity; 2.4 Paths; 2.5 The extended plane; stereographic projection 3. Fractions, powers, and roots: 3.1 Fractional linear transformations; 3.2 Properties of Mobius transformations; 3.3 Powers; 3.4 Roots; 3.5 The function $(z^2 +1)/(2z)$ 4. Holomorphic functions: 4.1 Complex-valued functions and continuity; 4.2 Differentiability; holomorphic functions; 4.3 The Cauchy-Riemann equations; 4.4 Laplace's equation; 4.5 The inverse function; 4.6 Conformal mapping; 4.7 Function spaces 5. Power series: 5.1 Infinite series; 5.2 Operations on series; 5.3 Double series; 5.4 Convergence of power series; 5.5 Power series as holomorphic functions; 5.6 Taylor's series; 5.7 Singularities; noncontinuable power series 6. Some elementary functions: 6.1 The exponential function; 6.2 The logarithm; 6.3 Arbitrary powers; the binomial series; 6.4 The trigonometric functions; 6.5 Inverse trigonometric functions 7. Complex integration: 7.1 Integration in the complex plane; 7.2 Cauchy's theorem; 7.3 Extensions; 7.4 Cauchy's integral; 7.5 Cauchy's formulas for the derivatives; 7.6 Integrals of the Cauchy type; 7.7 Analytic continuation: Schwarz's reflection principle; 7.8 The theorem of Morera; 7.9 The maximum principle; 7.10 Uniformly convergent sequences of holomorphic functions 8. Representation theorems: 8.1 Taylor's series; 8.2 The maximum modulus; 8.3 The Laurent expansion; 8.4 Isolated singularities; 8.5 Meromorphic functions; 8.6 Infinite products; 8.7 Entire functions; 8.8 The Gamma function 9. The calculus of residues: 9.1 The residue theorem; 9.2 The principle of the argument; 9.3 Summation and expansion theorems; 9.4 Inverse functions Appendix A: Some properties of point sets Appendix B: Some properties of polygons Appendix C: On the theory of integration Bibliography Index.

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