Analytic function theory

Einar Hille9780828402699, 0828402698

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.

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Table of contents :
Title ……Page 3
Foreword ……Page 5
Contents ……Page 7
Symbols ……Page 11
1.1. The real number system ……Page 13
1.2. Further properties of real numbers ……Page 20
1.3. The complex number system ……Page 25
2.1. Geometry of complex numbers ……Page 30
2.2. Curves and regions in the complex plane ……Page 38
2.3. Regions and convexity ……Page 41
2.4. Paths ……Page 45
2.5. The extended plane, stereographic projection ……Page 50
3.1. Fractional linear transformations ……Page 58
3.2. Properties of Mobius transformations ……Page 62
3.3. Powers ……Page 70
3.4. Roots ……Page 74
3.5. The function (z2 + l)/(2z) ……Page 77
4. HOLOMORPHIC FUNCTIONS ……Page 12
4.1. Complex-valued functions and continuity ……Page 80
4.2. Differentiability, holomorphic functions ……Page 84
4.3. The Cauchy-Riemann equations ……Page 90
4.4. Laplace’s equation ……Page 95
4.5. The inverse function ……Page 98
4.6. Conformal mapping ……Page 103
4.7. Function spaces ……Page 110
5.1. Infinite series ……Page 114
5.2. Operations on series ……Page 123
5.3. Double series ……Page 126
5.4. Convergence of power series ……Page 130
5.5. Power series as holomorphic functions ……Page 136
5.6. Taylor’s series ……Page 140
5.7. Singularities, noncontinuable power series ……Page 144
6.1. The exponential function ……Page 150
6.2. The logarithm ……Page 155
6.3. Arbitrary powers, the binomial series ……Page 159
6.4. The trigonometric functions ……Page 162
6.5. Inverse trigonometric functions ……Page 167
7.1. Integration in the complex plane ……Page 172
7.2. Cauchy’s theorem ……Page 175
7.3. Extensions ……Page 181
7.4. Cauchy’s integral ……Page 187
7.5. Cauchy’s formulas for the derivatives ……Page 190
7.6. Integrals of the Cauchy type ……Page 194
7.7. Analytic continuation: Schwarz’s reflection principle ……Page 196
7.8. The theorem of Morera ……Page 200
7.9. The maximum principle ……Page 201
7.10. Uniformly convergent sequences of holomorphic functions ……Page 203
8.1. Taylor’s series ……Page 208
8.2. The maximum modulus ……Page 214
8.3. The Laurent expansion ……Page 221
8.4. Isolated singularities ……Page 223
8.5. Merom orphic functions ……Page 229
8.6. Infinite products ……Page 234
8.7. Entire functions ……Page 237
8.8. The Gamma function ……Page 241
9.1. The residue theorem ……Page 253
9.2. The principle of the argument ……Page 264
9.3. Summation and expansion theorems ……Page 270
9.4. Inverse functions ……Page 277
Appendix A. Some Properties of Point Sets ……Page 289
B.l. The Jordan theorem ……Page 293
B.2. Triangulation ……Page 298
C.l. The Riemann integral ……Page 300
C.2. Functions of bounded variation ……Page 301
C.3. The Riemann-Stieltjes integral ……Page 304
Bibliography ……Page 309
Index ……Page 311