Items related to Theory of Complex Functions
Synopsis
Historical Introduction.- Chronological Table.- A. Elements of Function Theory.- 0. Complex Numbers and Continuous Functions.- §1. The field ? of complex numbers.- 1. The field ? - 2. ?-linear and ?-linear mappings ? ?? - 3. Scalar product and absolute value - 4. Angle-preserving mappings.- §2. Fundamental topological concepts.- 1. Metric spaces - 2. Open and closed sets - 3. Convergent sequences. Cluster points - 4. Historical remarks on the convergence concept - 5. Compact sets.- §3. Convergent sequences of complex numbers.- 1. Rules of calculation - 2. Cauchy's convergence criterion. Characterization of compact sets in ?.- §4. Convergent and absolutely convergent series.- 1. Convergent series of complex numbers - 2. Absolutely convergent series - 3. The rearrangement theorem - 4. Historical remarks on absolute convergence - 5. Remarks on Riemann's rearrangement theorem - 6. A theorem on products of series.- §5. Continuous functions.- 1. The continuity concept - 2. The ?-algebra C(X) - 3. Historical remarks on the concept of function - 4. Historical remarks on the concept of continuity.- §6. Connected spaces. Regions in ?.- 1. Locally constant functions. Connectedness concept - 2. Paths and path connectedness - 3. Regions in ? - 4. Connected components of domains - 5. Boundaries and distance to the boundary.- 1. Complex-Differential Calculus.- §1. Complex-differentiable functions.- 1. Complex-differentiability - 2. The Cauchy-Riemann differential equations - 3. Historical remarks on the Cauchy-Riemann differential equations.- §2. Complex and real differentiability.- 1. Characterization of complex-differentiable functions - 2. A sufficiency criterion for complex-differentiability - 3. Examples involving the Cauchy-Riemann equations - 4*. Harmonic functions.- §3. Holomorphic functions.- 1. Differentiation rules - 2. The C-algebra O(D) - 3. Characterization of locally constant functions - 4. Historical remarks on notation.- §4. Partial differentiation with respect to x, y, z and z.- 1. The partial derivatives fx, fy, fz, fz - 2. Relations among the derivatives ux, uy,Vx Vy, fx, fy, fz, fz - 3. The Cauchy-Riemann differential equation = 0 - 4. Calculus of the differential operators ? and ?.- 2. Holomorphy and Conformality. Biholomorphic Mappings...- §1. Holomorphic functions and angle-preserving mappings.- 1. Angle-preservation, holomorphy and anti-holomorphy - 2. Angle- and orientation-preservation, holomorphy - 3. Geometric significance of angle-preservation - 4. Two examples - 5. Historical remarks on conformality.- §2. Biholomorphic mappings.- 1. Complex 2×2 matrices and biholomorphic mappings - 2. The biholomorphic Cay ley mapping ? ?? - 3. Remarks on the Cay ley mapping - 4*. Bijective holomorphic mappings of ? and E onto the slit plane.- §3. Automorphisms of the upper half-plane and the unit disc.- 1. Automorphisms of ? - 2. Automorphisms of E - 3. The encryption for automorphisms of E - 4. Homogeneity of E and ?.- 3. Modes of Convergence in Function Theory.- §1. Uniform, locally uniform and compact convergence.- 1. Uniform convergence - 2. Locally uniform convergence - 3. Compact convergence - 4. On the history of uniform convergence - 5*. Compact and continuous convergence.- §2. Convergence criteria.- 1. Cauchy's convergence criterion - 2. Weierstrass' majorant criterion.- §3. Normal convergence of series.- 1. Normal convergence - 2. Discussion of normal convergence - 3. Historical remarks on normal convergence.- 4. Power Series.- §1. Convergence criteria.- 1. Abel's convergence lemma - 2. Radius of convergence - 3. The Cauchy-Hadamard formula - 4. Ratio criterion - 5. On the history of convergent power series.- §2. Examples of convergent power series.- 1. The exponential and trigonometric series. Euler's formula - 2. The logarithmic and arctangent series - 3. The binomial series - 4*. Convergence behavior on the boundary - 5 *. Abel's continuity theorem.- §3. Holomorphy of power series.- 1. F
"synopsis" may belong to another edition of this title.
- PublisherSpringer
- Publication date2011
- ISBN 10 1461209404
- ISBN 13 9781461209409
- BindingPaperback
- LanguageEnglish
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