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# Theory of complex functions

Author: Reinhold Remmert New York : Springer-Verlag, ©1991. Graduate texts in mathematics., Readings in mathematics ;, 122. Print book : EnglishView all editions and formats A lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. (not yet rated) 0 with reviews - Be the first.

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Material Type: Internet resource Book, Internet Resource Reinhold Remmert Find more information about: Reinhold Remmert 0387971955 9780387971957 3540971955 9783540971955 318419406 Funktionentheorie. 1. Pregrado xix, 453 p. : il. ; 25 cm. 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 2x2 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 differentiation and integration - 2. Holomorphy of power series. The interchange theorem - 3. Historical remarks on termwise differentiation of series - 4. Examples of holomorphic functions.- 4. 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.- 1. 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.- 2. 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.- 3. 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.- 4. 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.- 5. 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.- 0. 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.- 1. 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.- 2. Properties of complex path integrals.- 1. Rules of calculation - 2. The standard estimate - 3. Interchange theorems - 4. The integral ??B.- 3. 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.- 1. 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.- 2. 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.- 3. The development of holomorphic functions into power series.- 1. Lemma on developability - 2. The Cauchy-Taylor representation theorem - 3. Historical remarks on the representation theorem - 4. The Riemann continuation theorem - 5. Historical remarks on the Riemann continuation theorem.- 4. Discussion of the representation theorem.- 1. Holomorphy and complex-differentiability of every order - 2. The rearrangement theorem - 3. Analytic continuation - 4. The product theorem for power series - 5. Determination of radii of convergence.- 5 *. Special Taylor series. Bernoulli numbers.- 1. The Taylor series of z(ez - 1)-1. Bernoulli numbers - 2. The Taylor series of z cot z, tan z and - 3. Sums of powers and Bernoulli numbers - 4. Bernoulli polynomials.- C. Cauchy-Weierstrass-Riemann Function Theory.- 8. Fundamental Theorems about Holomorphic Functions.- 1. The Identity Theorem.- 1. The Identity Theorem - 2. On the history of the Identity Theorem - 3. Discreteness and countability of the a-places - 4. Order of a zero and multiplicity at a point - 5. Existence of singular points.- 2. The concept of holomorphy.- 1. Holomorphy, local integrability and convergent power series - 2. The holomorphy of integrals - 3. Holomorphy, angle- and orientation-preservation (final formulation) - 4. The Cauchy, Riemann and Weierstrass points of view. Weierstrass' creed.- 3. The Cauchy estimates and inequalities for Taylor coefficients.- 1. The Cauchy estimates for derivatives in discs - 2. The Gutzmer formula and the maximum principle - 3. Entire functions. LIOUVILLE' s theorem - 4. Historical remarks on the Cauchy inequalities and the theorem of Liouville - 5 *. Proof of the Cauchy inequalities following Weierstrass.- 4. Convergence theorems of Weierstrass.- 1. Weierstrass' convergence theorem - 2. Differentiation of series. Weierstrass' double series theorem - 3. On the history of the convergence theorems - 4. A convergence theorem for sequences of primitives - 5 *. A remark of Weierstrass' on holomorphy - 6 *. A construction of Weierstrass'.- 5. The open mapping theorem and the maximum principle.- 1. Open Mapping Theorem - 2. The maximum principle - 3. On the history of the maximum principle - 4. Sharpening the WEIERSTRASS convergence theorem - 5. The theorem of HURWITZ.- 9. Miscellany.- 1. The fundamental theorem of algebra.- 1. The fundamental theorem of algebra - 2. Four proofs of the fundamental theorem - 3. Theorem of Gauss about the location of the zeros of derivatives.- 2. Schwarz' lemma and the groups Aut E, Aut ?.- 1. Schwarz' lemma - 2. Automorphisms of E fixing 0. The groups Aut E and Aut ? - 3. Fixed points of automorphisms - 4. On the history of Schwarz' lemma - 5. Theorem of Study.- 3. Holomorphic logarithms and holomorphic roots.- 1. Logarithmic derivative. Existence lemma - 2. Homologically simply-connected domains. Existence of holomorphic logarithm functions - 3. Holomorphic root functions - 4. The equation $$f\left( z \right) = f\left( c \right)\exp \int {_{\gamma }\frac{{f'\left( \varsigma \right)}}{{f\left( \varsigma \right)}}} d\varsigma$$ 5. The power of square-roots.- 4. Biholomorphic mappings. Local normal forms.- 1. Biholomorphy criterion - 2. Local injectivity and locally biholomorphic mappings - 3. The local normal form - 4. Geometric interpretation of the local normal form - 5. Compositional factorization of holomorphic functions.- 5. General Cauchy theory.- 1. The index function ind?(z) - 2. The principal theorem of the Cauchy theory - 3. Proof of iii) ? ii) after DixON - 4. Nullhomology. Characterization of homologically simply-connected domains.- 6*. Asymptotic power series developments.- 1. Definition and elementary properties - 2. A sufficient condition for the existence of asymptotic developments - 3. Asymptotic developments and differentiation - 4. The theorem of Ritt - 5. Theorem of E. Borel.- 10. Isolated Singularities. Meromorphic Functions.- 1. Isolated singularities.- 1. Removable singularities. Poles - 2. Development of functions about poles - 3. Essential singularities. Theorem of Casorati and Weier-strass - 4. Historical remarks on the characterization of isolated singularities.- 2*. Automorphisms of punctured domains.- 1. Isolated singularities of holomorphic injections - 2. The groups Aut ? and Aut ?x - 3. Automorphisms of punctured bounded domains - 4. Conformally rigid regions.- 3. Meromorphic functions.- 1. Definition of meromorphy - 2. The C-algebra M(D) of the meromorphic functions in D - 3. Division of meromorphic functions - 4. The order function oc.- 11. Convergent Series of Meromorphic Functions.- 1. General convergence theory.- 1. Compact and normal convergence - 2. Rules of calculation - 3. Examples.- 2. The partial fraction development of ? cot ?z.- 1. The cotangent and its double-angle formula. The identity ? cot ?z = ?1(z) - 2. Historical remarks on the cotangent series and its proof - 3. Partial fraction series for . Characterizations of the cotangent by its addition theorem and by its differential equation.- 3. The Euler formulas for.- 1. Development of ?1(z) around 0 and Euler's formulas for ?(2n) - 2. Historical remarks on the Euler ?(2n)-formulas - 3. The differential equation for ?1 and an identity for the Bernoulli numbers - 4. The Eisenstein series.- 4*. The Eisenstein theory of the trigonometric functions.- 1. The addition theorem - 2. Eisenstein's basic formulas - 3. More Eisenstein formulas and the identity ?1 (z) = ? cot ?z - 4. Sketch of the theory of the circular functions according to Eisenstein.- 12. Laurent Series and Fourier Series.- 1. Holomorphic functions in annuli and Laurent series.- 1. Cauchy theory for annuli - 2. Laurent representation in annuli - 3. Laurent expansions - 4. Examples - 5. Historical remarks on the theorem of Laurent - 6*. Derivation of Laurent's theorem from the Cauchy-Taylor theorem.- 2. Properties of Laurent series.- 1. Convergence and identity theorems - 2. The Gutzmer formula and Cauchy inequalities - 3. Characterization of isolated singularities.- 3. Periodic holomorphic functions and Fourier series.- 1. Strips and annuli - 2. Periodic holomorphic functions in strips - 3. The Fourier development in strips - 4. Examples - 5. Historical remarks on Fourier series.- 4. The theta function.- 1. The convergence theorem - 2. Construction of doubly periodic functions - 3. The Fourier series of 4. Transformation formulas for the theta function - 5. Historical remarks on the theta function - 6. Concerning the error integral.- 13. The Residue Calculus.- 1. The residue theorem.- 1. Simply closed paths - 2. The residue - 3. Examples - 4. The residue theorem - 5. Historical remarks on the residue theorem.- 2. Consequences of the residue theorem.- 1. The integral 2. A counting formula for the zeros and poles - 3. Rouche's theorem.- 14. Definite Integrals and the Residue Calculus.- 1. Calculation of integrals.- 0. Improper integrals - 1. Trigonometric integrals - 2. Improper integrals 3. The integral for m, n ? ?, 0 < m < n.- 2. Further evaluation of integrals.- 1. Improper integrals 2. Improper integrals xa-1dx - 3. The integrals.- 3. Gauss sums.- 1. Estimation of 2. Calculation of the Gauss sums 3. Direct residue-theoretic proof of the formula 4. Fourier series of the Bernoulli polynomials.- Short Biographies o/Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass.- Photograph of Riemann's gravestone.- Literature.- Classical Literature on Function Theory - Textbooks on Function Theory - Literature on the History of Function Theory and of Mathematics Symbol Index.- Name Index.- Portraits of famous mathematicians 3. Graduate texts in mathematics., Readings in mathematics ;, 122. Reinhold Remmert ; translated by Robert B. Burckel.

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