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Analytic Function Theory, Volume I
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AMS Chelsea Publishing
1959; 308 pp; hardcover
Reprint/Revision History:
fifth printing with corrections 1982
ISBN-10: 0-8284-0269-8
ISBN-13: 978-0-8284-0269-9
List Price: US$45 Member Price: US$40.50
Order Code: CHEL/269

Second Edition.

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.

• 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 Möbius 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