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

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
3.4 - Rational Functions
3.5 - 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 - Mittag-Leffler’s theorem
4.13 - The general Mittag-Leffler problem
4.12 - The Weierstrass factorization of an integral function deduced from Mittag-Leffler’s theorem
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