Theory of Complex Functions

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.