Theory of Complex Functions

A lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations - in the original language with their English translation - from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure.

Inhalt

Historical Introduction.- Chronological Table.- A. Elements of Function Theory.- 0. Complex Numbers and Continuous Functions.- The field ? of complex numbers.- 1. The field ? - 2. ?-linear and ?-linear mappings ? ?? - 3. Scalar product and absolute value - 4. Angle-preserving mappings.- 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.- Convergent sequences of complex numbers.- 1. Rules of calculation - 2. Cauchy's convergence criterion. Characterization of compact sets in ?.- 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.- 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.- 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.- Complex-differentiable functions.- 1. Complex-differentiability - 2. The Cauchy-Riemann differential equations - 3. Historical remarks on the Cauchy-Riemann differential equations.- 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.- Holomorphic functions.- 1. Differentiation rules - 2. The C-algebra O(D) - 3. Characterization of locally constant functions - 4. Historical remarks on notation.- 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...- 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.- Biholomorphic mappings.- 1. Complex 2matrices 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.- 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.- 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.- Convergence criteria.- 1. Cauchy's convergence criterion - 2. Weierstrass' majorant criterion.- Normal convergence of series.- 1. Normal convergence - 2. Discussion of normal convergence - 3. Historical remarks on normal convergence.- 4. Power Series.- 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.- 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.- Holomorphy of power series.- 1. Formal term-wise differentiation and integration - 2. Holomorphy of power series. The interchange theorem - 3. Historical remarks on termwise differentiation of series - 4. Examples of holomorphic functions.- Structure of the algebra of convergent power series.- 1. The order function - 2. The theorem on units - 3. Normal form of a convergent power series - 4. Determination of all ideals.- 5. Elementary Transcendental Functions.- The exponential and trigonometric functions.- 1. Characterization of exp z by its differential equation - 2. The addition theorem of the exponential function - 3. Remarks on the addition theorem - 4. Addition theorems for cos z and sin z - 5. Historical remarks on cos z and sin z - 6. Hyperbolic functions.- The epimorphism theorem for exp z and its consequences.- 1. Epimorphism theorem - 2. The equation ker(exp) = 2?i? - 3. Periodicity of exp z - 4. Course of values, zeros, and periodicity of cos z and sin z - 5. Cotangent and tangent functions. Arctangent series - 6. The equation = i.- Polar coordinates, roots of unity and natural boundaries.- 1. Polar coordinates - 2. Roots of unity - 3. Singular points and natural boundaries - 4. Historical remarks about natural boundaries.- Logarithm functions.- 1. Definition and elementary properties - 2. Existence of logarithm functions - 3. The Euler sequence (1 + z/n)n - 4. Principal branch of the logarithm - 5. Historical remarks on logarithm functions in the complex domain.- Discussion of logarithm functions.- 1. On the identities log(wz) = log w + log z and log(exp z) = z - 2. Logarithm and arctangent - 3. Power series. The Newton-Abel formula - 4. The Riemann ?-function.- B. The Cauchy Theory.- 6. Complex Integral Calculus.- Integration over real intervals.- 1. The integral concept. Rules of calculation and the standard estimate - 2. The fundamental theorem of the differential and integral calculus.- Path integrals in ?.- 1. Continuous and piecewise continuously differentiable paths - 2. Integration along paths - 3. The integrals ??B(?-c)nb? - 4. On the history of integration in the complex plane - 5. Independence of parameterization - 6. Connection with real curvilinear integrals.- Properties of complex path integrals.- 1. Rules of calculation - 2. The standard estimate - 3. Interchange theorems - 4. The integral ??B.- Path independence of integrals. Primitives.- 1. Primitives - 2. Remarks about primitives. An integrability criterion - 3. Integrability criterion for star-shaped regions.- 7. The Integral Theorem, Integral Formula and Power Series Development.- The Cauchy Integral Theorem for star regions.- 1. Integral lemma of Goursat - 2. The Cauchy Integral Theorem for star regions - 3. On the history of the Integral Theorem - 4. On the history of the integral lemma - 5. Real analysis proof of the integral lemma - 6. The Presnel integrals cost2dt, sint2dt.- Cauchy's Integral Formula for discs.- 1. A sharper version of Cauchy's Integral Theorem for star regions - 2. The Cauchy Integral Formula for discs - 3. Historical remarks on the Integral Formula - 4. The Cauchy integral formula for continuously real-differentiable functions - 5*. Schwarz' integral formula.- The development of holomorphic functions into power series.…