Theory of complex functions / Reinhold Remmert ; translated by Robert B. Burckel.
Idioma: Inglés Lenguaje original: Alemán Series Graduate texts in mathematics: 122.; Graduate texts in mathematicsReadings in mathematics: Editor: New York : Springer-Verlag, c1991Descripción: xix, 453 p. : il. ; 25 cmISBN: 0387971955Títulos uniformes: Funktionentheorie. 1. Inglés Tema(s): Functions of complex variablesOtra clasificación: 30-01 Recursos en línea: Google Book SearchContents Preface to the English Edition v Preface to the Second German Edition vi Preface to the First German Edition vii Historical Introduction [1] Chronological Table [6] Part A. Elements of Function Theory Chapter 0. Complex Numbers and Continuous Functions [9] §1. The field C of complex numbers [10] 1. The field C —2. R-linear and C-linear mappings C → C — 3. Scalar product and absolute value — 4. Angle-preserving mappings §2. Fundamental topological concepts [17] 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 [22] 1. Rules of calculation — 2. Cauchy’s convergence criterion. Characterization of compact sets in C §4. Convergent and absolutely convergent series [26] 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 [34] 1. The continuity concept — 2. The C-algebra C(X) — 3. Historical remarks on the concept of function — 4. Historical remarks on the concept of continuity §6. Connected spaces. Regions in C [39] 1. Locally constant functions. Connectedness concept — 2. Paths and path connectedness — 3. Regions in C — 4. Connected components of domains — 5. Boundaries and distance to the boundary Chapter 1. Complex-Differential Calculus [45] §1. Complex-differentiable functions [47] 1. Complex-differentiability — 2. The Cauchy-Riemann differential equations — 3. Historical remarks on the Cauchy-Riemann differential equations §2. Complex and real differentiability [50] 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 [56] 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 [63] 1. The partial derivatives fx, fy, fz , f ͞z — 2. Relations among the derivatives Ux,Uy,Vx,Uy,fx,fy,fz,f͞z — 3. The Cauchy-Riemann differential equation (ϭf)/(ϭz) = [0] — 4. Calculus of the differential operators ϭ and ͞ϭ Chapter 2. Holomorphy and Conformality. Biholomorphic Mappings.. [71] §1. Holomorphic functions and angle-preserving mappings [72] 1. Angle-preservation, holomorphy and anti-holomorphy — 2. Angle- and orientation-preservation, holomorphy — 3. Geometric significance of anglepreservation — 4. Two examples — 5. Historical remarks on conformality §2. Biholomorphic mappings [80] 1. Complex 2x2 matrices and biholomorphic mappings — 2. The biholomorphic Cayley mapping H ͂→ E, z → (z-i)/z+i — 3. Remarks on the Cayley mapping — 4*. Bijective holomorphic mappings of H and E onto the slit plane §3. Automorphisms of the upper half-plane and the unit disc [85] 1. Automorphisms of H — 2. Automorphisms of E — 3. The encryption n(z-w/͞wz-1) automorphisms of E — 4. Homogeneity of E and H Chapter 3. Modes of Convergence in Function Theory [91] §1. Uniform, locally uniform and compact convergence [93] 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 [101] 1. Cauchy’s convergence criterion — 2. Weierstrass’ majorant criterion §3. Normal convergence of series [104] 1. Normal convergence — 2. Discussion of normal convergence — 3. Historical remarks on normal convergence Chapter J. Power Series [109] §1. Convergence criteria [110] 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 [115] 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 [123] 1. Formal term-wise differentiation and integration — 2. Holomorphy of power series. The interchange theorem — 3. Historical remarks on term-wise differentiation of series — 4. Examples of holomorphic functions §4. Structure of the algebra of convergent power series [128] 1. The order function — 2. The theorem on units — 3. Normal form of a convergent power series — 4. Determination of all ideals Chapter 5. Elementary Transcendental Functions [133] 1. The exponential and trigonometric functions [134] 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 [141] 1. Epimorphism theorem — 2. The equation ker(exp) = 2πiZ — 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 e i(π/2) = i 3. Polar coordinates, roots of unity and nat ural boundaries [148] 1. Polar coordinates — 2. Roots of unity — 3. Singular points and natural boundaries — 4. Historical remarks about natural boundaries Logarithm functions [154] 1, Definition and elementary properties — 2. Existence of logarithm functions — X 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 [160] 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 ς- function Part B. The Cauchy Theory Chapter 6. Comptex Integral Calculus [167] §0 Integration over real intervals [168] 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 C [171] 1. Continuous and piecewise continuously differentiable paths — 2. Integration along paths — 3. The integrals fϭB(C - c)ndς — 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 [178] 1. Rules of calculation — 2. The standard estimate — 3. Interchange theorems — 4. The integral 1/2πi ∫ϭB dς/ς-z §3. Path independence of integrals. Primitives [184] 1. Primitives — 2. Remarks about primitives. An integrability criterion — 3. Integrability criterion for star-shaped regions Chapter 7. The Integral Theorem, Integral Formula and Power Series Development [191] §1. The Cauchy Integral Theorem for star regions [192] 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 Fresnel integrals ∫0∞ cost2dt, ∫0∞ sin t2dt §2. Cauchy’s Integral Formula for discs [201] 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 [208] 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 [214] 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 [220] 1. The Taylor series of z(ez - l) -1. Bernoulli numbers — 2. The Taylor series of z cot z, tan z and z/sin z — 3. Sums of powers and Bernoulli numbers — 4. Bernoulli polynomials Part C. Cauchy-Weierstrass-Riemann Function Theory Chapter 8. Fundamental Theorems about Holomorphic Functions [227] §1. The Identity Theorem [227] 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 [236] 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 ... [241] 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 [248] 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 [256] 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 Chapter 9. Miscellany [265] §1. The fundamental theorem of algebra [265] 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 H [269] 1. Schwarz’ lemma — 2. Automorphisms of E fixing 0. The groups Aut E and Aut H — 3. Fixed points of automorphisms — 4. On the history of Schwarz’ lemma — 5. Theorem of Study §3. Holomorphic logarithms and holomorphic roots [276] 1. Logarithmic derivative. Existence lemma — 2. Homologically simply-connected domains. Existence of holomorphic logarithm functions — 3. Holomorphic root functions — 4. The equation f(z) = f(c) exp ∫ϓ (f'(ς)/f(ς))dς — 5. The power of square-roots §4. Biholomorphic mappings. Local normal forms [281] 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 [287] 1. The index function indϓ(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 [293] 1. Definition and elementary properties — 2. A sufficient condition for the existence of asymptotic developments — 3. Asvmptotic developments and differentiation — 4. The theorem of Ritt — 5. Theorem of É. Borel Chapter 10. Isolated Singularities. Meromorphic Functions [303] §1. Isolated singularities [303] 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 [310] 1. Isolated singularities of holomorphic injections — 2. The groups Aut C and AutCx — 3. Automorphisms of punctured bounded domains — 4. Conformally rigid regions §3. Meromorphic functions [315] 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 Chapter 11. Convergent Series of Meromorphic Functions [321] §1. General convergence theory [321] 1. Compact and normal convergence — 2. Rules of calculation — 3. Examples §2. The partial fraction development of π cot π z [325] 1. The cotangent and its double-angle formula. The identity π cot π z = ϵ1(z) — 2. Historical remarks on the cotangent series and its proof — 3. Partial fraction series for (π2)/(sin2 πz) and (π/sin πz) — 4*. Characterizations of the cotangent by its addition theorem and by its differential equation §3. The Euler formulas for Σ v ≥1 v-2n [331] 1. Development of ϵ1(z) around 0 and Euler’s formulas for ς(2n) — 2. Historical remarks on the Euler ς(2n)-formulas — 3. The differential equation for ϵ1 and an identity for the Bernoulli numbers — 4. The Eisenstein series ϵk(z):= Σ∞-∞ 1/(z+v)k §4*. The Eisenstein theory of the trigonometric functions [335] 1. The addition theorem — 2. Eisenstein’s basic formulas — 3. More Eisenstein formulas and the identity ϵ1(z) = π cot π z — 4. Sketch of the theory of the circular functions according to Eisenstein Chapter 12. Laurent Series and Fourier Series [343] §1. Holomorphic functions in annuli and Laurent series [343] 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 [356] 1. Convergence and identity theorems — 2. The Gutzmer formula and Cauchy inequalities — 3. Characterization of isolated singularities §3. Periodic holomorphic functions and Fourier series [361] 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 [365] 1. The convergence theorem — 2. Construction of doubly periodic functions — 3. The Fourier series of e-z2πT ϴ(iTz, T) — 4. Transformation formulas for the theta function — 5. Historical remarks on the theta function — 6. Concerning the error integral Chapter 13. The Residue Calculus [377] §1. The residue theorem [377] 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 [387] 1. The integral 1/(2πi) ∫ϓ F(ς) [(f'(ς))/(f(ς)-a)] dς — 2. A counting formula for the zeros and poles — 3, Rouche’s theorem Chapter 14. Definite Integrals and the Residue Calculus [395] §1. Calculation of integrals [395] 0. Improper integrals — 1. Trigonometric integrals ∫2π0 R(cosℓ, sin ℓ)dℓ — 2. Improper integrals ∫∞-∞ f(x)dx — 3. The integral ∫0∞ (xm-1)/(1+xn ) for m,n ϵ N, 0<m<n §2. Further evaluation of integrals [401] 1. Improper integrals ∫∞-∞ g(x)eiaxdx — 2. Improper integrals f0∞ q(x) xa~1dx — 3. The integrals f0∞ (sinn x/xn)dx §3. Gauss sums [409] 1. Estimation of euz/ez-1 for 0 ≤ u ≤ [1] — 2. Calculation of the Gauss sums Gn := ∑ 0n-1 e 2(πi/n)v2, n ≥ [1] — 3. Direct residue-theoretic proof of the formula ∫∞-∞e~t2dt = √π — 4. Fourier series of the Bernoulli polynomials Short Biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass [417] Photograph of Riemann’s gravestone [422] Literature [423] Classical Literature on Function Theory — Textbooks on Function Theory — Literature on the History of Function Theory and of Mathematics Symbol Index [435] Name Index [437] Subject Index [443] Portraits of famous mathematicians 3, [341]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 30 R388 (Browse shelf) | Checked out | 2024-02-05 | A-6692 |
Traducción de: Funktionentheorie I. 2nd ed. Springer, 1989.
Incluye referencias bibliográficas (p. 423-434) e índices.
Contents --
Preface to the English Edition v --
Preface to the Second German Edition vi --
Preface to the First German Edition vii --
Historical Introduction [1] --
Chronological Table [6] --
Part A. Elements of Function Theory --
Chapter 0. Complex Numbers and Continuous Functions [9] --
§1. The field C of complex numbers [10] --
1. The field C --
—2. R-linear and C-linear mappings C → C --
— 3. Scalar product and absolute value --
— 4. Angle-preserving mappings --
§2. Fundamental topological concepts [17] --
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 [22] --
1. Rules of calculation --
— 2. Cauchy’s convergence criterion. Characterization of compact sets in C --
§4. Convergent and absolutely convergent series [26] --
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 [34] --
1. The continuity concept --
— 2. The C-algebra C(X) --
— 3. Historical remarks on the concept of function --
— 4. Historical remarks on the concept of continuity --
§6. Connected spaces. Regions in C [39] --
1. Locally constant functions. Connectedness concept --
— 2. Paths and path connectedness --
— 3. Regions in C --
— 4. Connected components of domains --
— 5. Boundaries and distance to the boundary --
Chapter 1. Complex-Differential Calculus [45] --
§1. Complex-differentiable functions [47] --
1. Complex-differentiability --
— 2. The Cauchy-Riemann differential equations --
— 3. Historical remarks on the Cauchy-Riemann differential equations --
§2. Complex and real differentiability [50] --
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 [56] --
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 [63] --
1. The partial derivatives fx, fy, fz , f ͞z --
— 2. Relations among the derivatives Ux,Uy,Vx,Uy,fx,fy,fz,f͞z --
— 3. The Cauchy-Riemann differential equation (ϭf)/(ϭz) = [0] --
— 4. Calculus of the differential operators ϭ and ͞ϭ --
Chapter 2. Holomorphy and Conformality. Biholomorphic Mappings.. [71] --
§1. Holomorphic functions and angle-preserving mappings [72] --
1. Angle-preservation, holomorphy and anti-holomorphy --
— 2. Angle- and orientation-preservation, holomorphy --
— 3. Geometric significance of anglepreservation --
— 4. Two examples — 5. Historical remarks on conformality --
§2. Biholomorphic mappings [80] --
1. Complex 2x2 matrices and biholomorphic mappings --
— 2. The biholomorphic Cayley mapping H ͂→ E, z → (z-i)/z+i --
— 3. Remarks on the Cayley mapping --
— 4*. Bijective holomorphic mappings of H and E onto the slit plane --
§3. Automorphisms of the upper half-plane and the unit disc [85] --
1. Automorphisms of H --
— 2. Automorphisms of E --
— 3. The encryption n(z-w/͞wz-1) automorphisms of E --
— 4. Homogeneity of E and H --
Chapter 3. Modes of Convergence in Function Theory [91] --
§1. Uniform, locally uniform and compact convergence [93] --
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 [101] --
1. Cauchy’s convergence criterion --
— 2. Weierstrass’ majorant criterion --
§3. Normal convergence of series [104] --
1. Normal convergence --
— 2. Discussion of normal convergence --
— 3. Historical remarks on normal convergence --
Chapter J. Power Series [109] --
§1. Convergence criteria [110] --
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 [115] --
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 [123] --
1. Formal term-wise differentiation and integration --
— 2. Holomorphy of power series. The interchange theorem --
— 3. Historical remarks on term-wise differentiation of series --
— 4. Examples of holomorphic functions --
§4. Structure of the algebra of convergent power series [128] --
1. The order function --
— 2. The theorem on units --
— 3. Normal form of a convergent power series --
— 4. Determination of all ideals --
Chapter 5. Elementary Transcendental Functions [133] --
1. The exponential and trigonometric functions [134] --
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 [141] --
1. Epimorphism theorem --
— 2. The equation ker(exp) = 2πiZ --
— 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 e i(π/2) = i --
3. Polar coordinates, roots of unity and nat ural boundaries [148] --
1. Polar coordinates --
— 2. Roots of unity --
— 3. Singular points and natural boundaries --
— 4. Historical remarks about natural boundaries --
Logarithm functions [154] --
1, Definition and elementary properties --
— 2. Existence of logarithm functions --
— X 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 [160] --
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 ς- function --
Part B. The Cauchy Theory --
Chapter 6. Comptex Integral Calculus [167] --
§0 Integration over real intervals [168] --
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 C [171] --
1. Continuous and piecewise continuously differentiable paths --
— 2. Integration along paths --
— 3. The integrals fϭB(C - c)ndς --
— 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 [178] --
1. Rules of calculation --
— 2. The standard estimate --
— 3. Interchange theorems --
— 4. The integral 1/2πi ∫ϭB dς/ς-z --
§3. Path independence of integrals. Primitives [184] --
1. Primitives --
— 2. Remarks about primitives. An integrability criterion --
— 3. Integrability criterion for star-shaped regions --
Chapter 7. The Integral Theorem, Integral Formula and Power Series Development [191] --
§1. The Cauchy Integral Theorem for star regions [192] --
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 Fresnel integrals ∫0∞ cost2dt, ∫0∞ sin t2dt --
§2. Cauchy’s Integral Formula for discs [201] --
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 [208] --
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 [214] --
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 [220] --
1. The Taylor series of z(ez - l) --
-1. Bernoulli numbers --
— 2. The Taylor series of z cot z, tan z and z/sin z --
— 3. Sums of powers and Bernoulli numbers --
— 4. Bernoulli polynomials Part C. Cauchy-Weierstrass-Riemann Function Theory --
Chapter 8. Fundamental Theorems about Holomorphic Functions [227] --
§1. The Identity Theorem [227] --
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 [236] --
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 ... [241] --
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 [248] --
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 [256] --
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 --
Chapter 9. Miscellany [265] --
§1. The fundamental theorem of algebra [265] --
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 H [269] --
1. Schwarz’ lemma --
— 2. Automorphisms of E fixing 0. The groups Aut E and Aut H --
— 3. Fixed points of automorphisms --
— 4. On the history of Schwarz’ lemma --
— 5. Theorem of Study --
§3. Holomorphic logarithms and holomorphic roots [276] --
1. Logarithmic derivative. Existence lemma --
— 2. Homologically simply-connected domains. Existence of holomorphic logarithm functions --
— 3. Holomorphic root functions --
— 4. The equation f(z) = f(c) exp ∫ϓ (f'(ς)/f(ς))dς --
— 5. The power of square-roots --
§4. Biholomorphic mappings. Local normal forms [281] --
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 [287] --
1. The index function indϓ(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 [293] --
1. Definition and elementary properties --
— 2. A sufficient condition for the existence of asymptotic developments --
— 3. Asvmptotic developments and differentiation --
— 4. The theorem of Ritt --
— 5. Theorem of É. Borel --
Chapter 10. Isolated Singularities. Meromorphic Functions [303] --
§1. Isolated singularities [303] --
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 [310] --
1. Isolated singularities of holomorphic injections --
— 2. The groups Aut C and AutCx --
— 3. Automorphisms of punctured bounded domains --
— 4. Conformally rigid regions --
§3. Meromorphic functions [315] --
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 --
Chapter 11. Convergent Series of Meromorphic Functions [321] --
§1. General convergence theory [321] --
1. Compact and normal convergence --
— 2. Rules of calculation --
— 3. Examples --
§2. The partial fraction development of π cot π z [325] --
1. The cotangent and its double-angle formula. The identity π cot π z = ϵ1(z) --
— 2. Historical remarks on the cotangent series and its proof --
— 3. Partial fraction series for (π2)/(sin2 πz) and (π/sin πz) --
— 4*. Characterizations of the cotangent by its addition theorem and by its differential equation --
§3. The Euler formulas for Σ v ≥1 v-2n [331] --
1. Development of ϵ1(z) around 0 and Euler’s formulas for ς(2n) --
— 2. Historical remarks on the Euler ς(2n)-formulas --
— 3. The differential equation for ϵ1 and an identity for the Bernoulli numbers --
— 4. The Eisenstein series ϵk(z):= Σ∞-∞ 1/(z+v)k --
§4*. The Eisenstein theory of the trigonometric functions [335] --
1. The addition theorem --
— 2. Eisenstein’s basic formulas --
— 3. More Eisenstein formulas and the identity ϵ1(z) = π cot π z --
— 4. Sketch of the theory of the circular functions according to Eisenstein --
Chapter 12. Laurent Series and Fourier Series [343] --
§1. Holomorphic functions in annuli and Laurent series [343] --
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 [356] --
1. Convergence and identity theorems --
— 2. The Gutzmer formula and Cauchy inequalities --
— 3. Characterization of isolated singularities --
§3. Periodic holomorphic functions and Fourier series [361] --
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 [365] --
1. The convergence theorem --
— 2. Construction of doubly periodic functions --
— 3. The Fourier series of e-z2πT ϴ(iTz, T) --
— 4. Transformation formulas for the theta function --
— 5. Historical remarks on the theta function --
— 6. Concerning the error integral --
Chapter 13. The Residue Calculus [377] --
§1. The residue theorem [377] --
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 [387] --
1. The integral 1/(2πi) ∫ϓ F(ς) [(f'(ς))/(f(ς)-a)] dς --
— 2. A counting formula for the zeros and poles --
— 3, Rouche’s theorem --
Chapter 14. Definite Integrals and the Residue Calculus [395] --
§1. Calculation of integrals [395] --
0. Improper integrals --
— 1. Trigonometric integrals ∫2π0 R(cosℓ, sin ℓ)dℓ --
— 2. Improper integrals ∫∞-∞ f(x)dx --
— 3. The integral ∫0∞ (xm-1)/(1+xn ) for m,n ϵ N, 0<m<n --
§2. Further evaluation of integrals [401] --
1. Improper integrals ∫∞-∞ g(x)eiaxdx --
— 2. Improper integrals f0∞ q(x) xa~1dx --
— 3. The integrals f0∞ (sinn x/xn)dx --
§3. Gauss sums [409] --
1. Estimation of euz/ez-1 for 0 ≤ u ≤ [1] --
— 2. Calculation of the Gauss sums Gn := ∑ 0n-1 e 2(πi/n)v2, n ≥ [1] --
— 3. Direct residue-theoretic proof of the formula ∫∞-∞e~t2dt = √π --
— 4. Fourier series of the Bernoulli polynomials --
Short Biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass [417] --
Photograph of Riemann’s gravestone [422] --
Literature [423] --
Classical Literature on Function Theory — Textbooks on Function Theory — Literature on the History of Function Theory and of Mathematics --
Symbol Index [435] --
Name Index [437] --
Subject Index [443] --
Portraits of famous mathematicians 3, [341] --
MR, 91m:30001
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