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Mathematics - Analysis | Theory of Complex Functions

Theory of Complex Functions

Remmert, Reinhold

Translated by Burckel, R.B.

1991, XIX, 458 p.

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Wohl dem, der seiner Viiter gem gedenkt (Blessings on him who gladly remembers his forefathers) - J. W. v. GOETHE 1. . . . "Zuvorderst wiirde ich jemand, der eine neue Function in die Analyse einfiihren will, urn eine Erklarung bitten, ob er sie schlechterdings bloss auf reelle Grossen (reelle Werthe des Arguments der Function) angewandt wis­ sen will, und die imaginaren Wert he des Arguments gleichsam nur als ein Uberbein ansieht - oder ob er meinem Grundsatz beitrete, dass man in dem Reiche der Grossen die imaginaren a + byCI = a + bi als gleiche Rechte mit den reellen geniessend ansehen miisse. Es ist hier nicht von prakti­ schem Nutzen die Rede, sondern die Analyse ist mir eine selbstandige Wis­ senschaft, die durch Zuriicksetzung jener fingirten Grossen ausserordentlich an Schonheit und Rundung verlieren und alle Augenblick Wahrheiten, die sonst allgemein gelten, hochst liiBtige Beschrankungen beizufiigen genothigt sein wiirde . . . (At the very beginning I would ask anyone who wants to introduce a new function into analysis to clarify whether he intends to confine it to real magnitudes (real values of its argument) and regard the imaginary values as just vestigial - or whether he subscribes to my fun­ damental proposition that in the realm of magnitudes the imaginary ones a + byCI = a + bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analy­ sis is, to my mind, a self-sufficient science.

Content Level » Graduate

Keywords » Fourier series - Meromorphic function - calculus - convergence - differential calculus - holomorphic function - integral - residue

Related subjects » Analysis

Table of contents 

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. 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 É. 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. Rouché’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.

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