Lectures on the Theory of Functions of a Complex Variable, Vol. 1: Holomorphic Functions

Table of contents :
Title
Chapter 1 HOLOMORPHIC FUNCTIONSPOWER SERIES AS HOLOMORPHIC FUNCTIONS ELEMENTARY FUNCTIONS
1.1 The complex plane
1.2 Continuous functions
1.3 Holomorphic functions
1.4 Conjugate functions
1.5 Sequences of functions
1.6 Power Series
1.7 The power series as a holomorphic function
1.8 The theorems of Picard and Abel
1.9 The A-summability of a series. Tauber’s theorem
1.10 The exponential, circular and hyperbolic functions
1.11 The logarithm and the power
1.12 The inverse circular and hyperbolic functions
Chapter 2
CAUCHY’S INTEGRAL THEOREM AND
ITS COROLLARIES - EXPANSION IN TAYLOR SERIES
2.1 Chains and cycles
2.2 The connectivity of a region
2.3 The line integral of a complex function
2.4 Properties of the line integrals of complex functions
2.5 Cauchy’s integral theorem
2.6 The fundamental theorem of algebra
2.7 Cauchy’s integral formula
2.8 Formula for the derivative. Riemann’s theorem
2.9 Differentiation inside the sign of integration
2.10 Morera’s theorem
2.11 Zeros of a holomorphic function
2.12 The Cauchy-Liouville theorem
2.13 The maximum modulus theorem
2.14 Real parts of holomorphic functions
2.15 Representation of a holomorphic function by its real part
2.16 The Taylor expansion
2.17 Some remarkable power series expansions
2.18 Cauchy’s inequality. Parseval’s identity
2.19 An extension of the Cauchy-Liouville theorem
2.20 Weierstrass’s theorems about the limits of sequences of functions
2.21 Schwarz’s lemma
2.22 Vitali’s theorem
2.23 Laurent’s expansion. The Fourier series
Chapter 3 REGULAR AND SINGULAR POINTS-RESIDUES-ZEROS
3.1 Regular points
3.2 Isolated singularities
3.3 Residues
3.4 Rational Functions
3.5 The theorem of residues
3.6 Evaluation of some integrals by means of the theorem of residues
3.7 Evaluation of the sum of certain series
3.8 The logarithmic derivative
3.9 Jensen’s theorem. The Poisson-Jensen formula
3.10 Rouche’s theorem
3.11 A theorem of Hurwitz
3.12 The mapping of a region
3.13 Generalization of Taylor’s and Laurent’s series
3.14 Legendre’s polynomials
Chapter 4 WEIERSTRASS’S FACTORIZATION OF INTEGRAL FUNCTIONS - CAUCHY’S EXPANSION OF PARTIAL FRACTIONS - MITTAG-LEFFLER’S PROBLEM
4.1 Infinite products
4.2 The factorization of integral functions
4.3 Primary factors of Weierstrass
4.4 Expansion of an integral function in an infinite product
4.5 Canonical products
4.6 The gamma function
4.7 The Eulerian integrals
4.8 The Gaussian psi function
4.9 Binet’s function
4.10 Cauchy’s method for the decomposition of mero­
morphic functions into partial fractions
4.11 The Mittag-Leffler problem
4.12 The Weierstrass factorization of an integral function deduced from Mittag-Leffler’s theorem
4.13 The general Mittag-Leffler problem
Chapter 5 ELLIPTIC FUNCTIONS
5.1 Periodic functions
5.2 Elliptic functions
5.3 The pe function of Weierstrass
5.4 The differential equation of the pe function
5.5 Addition theorems
5.6 The sigma functions of Weierstrass
5.7 The bisection formula of the pe function
5.8 The theta functions of Jacobi
5.9 The expression for the theta functions as infinite products
5.10 Jacobi’s imaginary transformation
5.11 The logarithmic derivative of the theta functions
5.12 The pe function with real invariants
5.13 The periods represented as integrals
5.14 The Jacobian elliptic functions
5.15 Fourier expansions of the Jacobian functions
5.16 Addition theorems
5.17 Legendre’s elliptic integral of the second kind
Chapter 6 INTEGRAL FUNCTIONS OF FINITE ORDER
6.1 The genus of an integral function
6.2 The theorems of Laguerre
6.3 Poincare’s theorems
6.4 The order of an integral function
6.5 Integral functions with a finite number of zeros
6.6 The order of a function related to the coefficients of its Taylor expansion
6.7 Hadamard’s first theorem
6.8 Hadamard’s second theorem
6.9 Hadamard’s factorization theorem
6.10 The Borel-Caratheodory theorem
6.11 Picard’s theorem for integral functions of finite order
6.12 The theorem of Phragmen
6.13 Mittag-Leffler’s function
Chapter 7 DIRICHLET SERIES THE ZETA FUNCTION OF RIEMANN THE LAPLACE INTEGRAL
7.1 Dirichlet series. Absolute convergence
7.2 Simple convergence
7.3 Formulas for the abscissa of convergence
7.4 The representation of a Dirichlet series by an infinite integral
7.5 The functional equation of the zeta function
7.6 Euler’s infinite product
7.7 Some properties of the zeta function
7.8 The existence of zeros in the critical strip
7.9 The generalized zeta function
7.10 The representation of the generalized zeta function by a loop integral
7.11 Perron’s formula
7.12 A formula of Hadamard
7.13 Representation of the sum of a Dirichlet series as a Laplace integral
7.14 The Laplace integral
7.15 Abscissa of convergence
7.16 Regularity
7.17 Some remarkable integrals of the Laplace type
7.18 The prime number theorem
7.19 The incomplete gamma functions
7.20 Representability of a function as a Laplace integral
Chapter 8 SUMMABILITY OF POWER SERIES OUTSIDE THE CIRCLE OF CONVERGENCE SUM FORMULAS - ASYMPTOTIC SERIES
8.1 The principal star of a function
8.2 Existence of barrier-points
8.3 The Borel summability of a power series
8.4 The Mittag-Leffler summability of a power series
8.5 Plana’s sum formula
8.6 The Euler-Maclaurin sum formula
8.7 Stirling’s series
8.8 The Bernoullian polynomials
8.9 The associate periodic functions
8.10 Asymptotic expansions
8.11 Asymptotic expansion of Laplace integrals
8.12 Illustrative examples
8.13 Rotation of the path of integration
8.14 The method of steepest descents
INDEX