Sinnvoll24
Theory of Complex Functions
Graduate Texts in Mathematics 122 - Readings in Mathematics
Langbeschreibung
RezensionR. Remmert and R.B. Burckel Theory of Complex Functions "Its accessibility makes it very useful for a first graduate course on complex function theory, especially where there is an opportunity for developing an interest on the part of motivated students in the history of the subject. Historical remarks abound throughout the text. Short biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass are given. There is an extensive bibliography of classical works on complex function theory with comments on some of them. In addition, a list of modern complex function theory texts and books on the history of the subject and of mathematics is given. Throughout the book there are numerous interesting quotations. In brief, the book affords splendid opportunities for a rich treatment of the subject."-MATHEMATICAL REVIEWS
Hauptbeschreibung
Covers material from function theory up to residue calculus, including discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations from their classical works. Includes many examples and practice exercises. Annotation copyright Book News, Inc. Portland, Or.
Inhaltsverzeichnis
Historical Introduction.- Chronological Table.- Elements of Function Theory.- The Cauchy Theory.- Cauchy-Weierstrass-Riemann Function Theory.- Short Biographies.- Literature.- Indices.
InhaltsangabeHistorical 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. Formal term-wise differentia
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. Formal term-wise differentia
Autor*in:
Reinhold Remmert
Art:
Gebunden/Hardback
Sprache :
Englisch
ISBN-13:
9780387971957
Verlag:
Springer Verlag GmbH
Erscheinungsdatum:
30.04.1990
Erscheinungsjahr:
1990
Ausgabe:
1/1999
Maße:
24.2x16.2x3 cm
Seiten:
458
Gewicht:
858 g