Applied and Computational Complex Analysis - Volume 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions

✍ Scribed by Peter Henrici

• Publisher: Wiley
• Year: 1977
• Tongue: English
• Leaves: 678
• Edition: 1
• Category: Library

✦ Synopsis

Presents applications as well as the basic theory of analytic functions of one or several complex variables. The first volume discusses applications and basic theory of conformal mapping and the solution of algebraic and transcendental equations. Volume Two covers topics broadly connected with ordinary differental equations: special functions, integral transforms, asymptotics and continued fractions. Volume Three details discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions, potential theory in the plane and polynomial expansions.

✦ Table of Contents

8 Infinite Products
8.1. Definition and Elementary Properties
8.2. Some Infinite Products Relevant to Number Theory
8.3. Product Representations of Entire Functions
8.4. The Gamma Function
8.5. Stirling's Formula
8.6. Some Special Series and Products
8.7. The Beta Function
8.8. Integrals of the Mellin-Barnes Type
Seminar Assignments
Notes
9 Ordinary Differential Equations
9.1. The Existence Theorem
9.2. Power Series Method
9.3. Linear Systems
9.4. Linear Systems with Isolated Singularities
9.5. Singularities of the First Kind: Formal Solutions
9.6. Scalar Equations of Higher Order: Method of Frobenius
9.7. Two Examples: The Equations of Kummer and Bessel
9.8. The Infinite Point: Equations of Fuchsian Type
9.9. The Hypergeometric Differential Equation
9.10. Quadratic Transforms: Legendre Functions
9.11. Singularities of the Second Kind: Formal Solutions
9.12. Singularities of the Second Kind of Special Second-Order Equations
Seminar Assignments
Notes
10 Integral Transforms
10.1. Definition and Basic Properties of the:?? Transformation
10.2. Operational Rules: Basic Correspondences
10.3. Ordinary Differential Equations: Systems
10.4. Convolution
10.5. Some Nonelementary Correspondences
10.6. The Fourier Integral
10.7. The Laplace Transform as a Fourier Transform
10.8. Dirichlet Series: Prime Number Theorem
10.9. Functions of Exponential Type
10.10. The Discrete Laplace Transform
10.11. Some Integral Transforms Related to the ?? Transform
10.12. Some Applications to Partial Differential Equations
Seminar Assignments
Notes
11 Asymptotic Methods
11.1. An Example: Asymptotic Power Series
11.2. The Algebra of Asymptotic Power Series
11.3. Analytic Properties of Asymptotic Power Series
11.4. Asymptotic Solutions of Differential Equations
11.5. The Watson-Doetsch Lemma
11.6. Extension of the Lemma
11.7. Asymptotic Formulas: Laplace's Method
11.8. The Method of Steepest Descent
11.9. General Asymptotic Expansions: Asymptotic Factorial Series
11.10. Generating Functions: Subtracted Singularities
11.11. The Euler-Maclaurin Summation Formula
11.12. The Numerical Evaluation of Limits: Romberg's Algorithm
Seminar Assignments
Notes
12 Continued Fractions
12.1. Definition and Basic Properties
12.2. Continued Fractions in Number Theory
12.3. Convergence of Continued Fractions with Complex Elements
12.4. RITZ Fractions (Formal Theory): Pade Table
12.5. The Convergence of RITZ Fractions: Examples
12.6. Tne Division Algorithm: Rational RITZ Fractions
12.7. SITZ Fractions: Approximants, Stable Polynomials
12.8. S Fractions: Generalized Value Functions, Convergence
12.9. S Fractions: The Representation of Tneir Generalized Value Functions by Stieltjes Transforms
12.10. Positive Symmetric Functions and Their Representation as Stieltjes Transforms
12.11. Existence and Convergence of the S Fraction Corresponding to a Stieltjes Transform
12.12. S Fractions: Expansions of Stieltjes Transforms
12.13. S Fractions: Expansions of Iterated Laplace Transforms
12.14. Moment Problems
Seminar Assignments
Notes
Bibliography
Appendix: Some additional problems on vol. I
Index