Selected Asymptotic Methods with Applications to Electromagnetics and Antennas

Book description

This book describes and illustrates the application of several asymptotic methods that have proved useful in the authors' research in electromagnetics and antennas. We first define asymptotic approximations and expansions and explain these concepts in detail. We then develop certain prerequisites from complex analysis such as power series, multivalued functions (including the concepts of branch points and branch cuts), and the all-important gamma function. Of particular importance is the idea of analytic continuation (of functions of a single complex variable); our discussions here include some recent, direct applications to antennas and computational electromagnetics. Then, specific methods are discussed. These include integration by parts and the Riemann-Lebesgue lemma, the use of contour integration in conjunction with other methods, techniques related to Laplace's method and Watson's lemma, the asymptotic behavior of certain Fourier sine and cosine transforms, and the Poisson summation formula (including its version for finite sums). Often underutilized in the literature are asymptotic techniques based on the Mellin transform; our treatment of this subject complements the techniques presented in our recent Synthesis Lecture on the exact (not asymptotic) evaluation of integrals.

Table of contents

  1. Preface
    1. References
  2. Introduction: Simple Asymptotic Approximations
    1. Far Field of Linear Antenna
    2. Period of Simple Pendulum: Small Oscillations
    3. A Differential Equation
      1. ``Series Solution''
      2. The Solution as an Integral: Integration by Parts
    4. Asymptotic Approximations for High-SWR Transmission Lines
      1. Exact Formulas
      2. Further Exact Formulas: Large- and Small- g regions
      3. Asymptotic Formulas for the Large-g Region
      4. Asymptotic Formulas for the Small-g Region
    5. Supplementary Remarks and Further Reading
    6. Problems
    7. References
  3. Asymptotic Approximations Defined
    1. Definitions
    2. Remarks and Examples
    3. Compound Asymptotic Approximations
      1. Elementary Example
      2. Bessel Function of Order Zero
    4. Asymptotic Expansions
    5. Historical and Supplementary Remarks
    6. Problems
    7. References
  4. Concepts from Complex Variables
    1. Gamma Function and Related Functions
    2. Power Series
    3. Analytic Continuation
      1. Removable Singularities
      2. Geometric Series
      3. Analytic Continuation Defined: Uniqueness
      4. Further Examples
      5. Integrals that are Analytic Functions of a Parameter
    4. Multivalued Functions and Branch Points
      1. Square Root
      2. Further Examples
      3. The Point at Infinity
    5. Branches and Principal Values of Multivalued Functions
      1. Square Root
      2. Logarithm and Powers Other Than the Square Root
      3. The function z2-1
      4. Values on Branch Cuts
    6. Applications to Antennas and Electromagnetics: Nonsolvability
      1. Hallén's and Pocklington's Equations with the Approximate Kernel
      2. Integral Equation Related to the Method of Auxiliary Sources (MAS)
    7. Supplementary Remarks and Further Reading
    8. Problems (1/2)
    9. Problems (2/2)
    10. References
  5. Laplace's Method and Watson's Lemma
    1. Laplace's Method
      1. Simple Example
      2. Related Examples
      3. Stirling's Formula: Leading Term
      4. An Application to the Thin-Wire Circular-Loop Antenna
    2. Watson's Lemma
      1. Statement of Lemma and Motivation
      2. Remarks and Extensions
      3. Examples
      4. Stirling's Formula Revisited and Lagrange Inversion Theorem
      5. An Application to the Method of Auxiliary Sources
    3. Additional Remarks
    4. Problems
    5. References
  6. Integration by Parts and Asymptotics of Some Fourier Transforms
    1. Integration by Parts and Laplace Transforms
      1. Complementary Error Function
      2. Remarks
    2. Integration by Parts and Fourier Transforms
      1. Simple Example: Riemann-Lebesgue Lemma
      2. Remarks on the Lemma
      3. Simple Example Continued
      4. Example with Zero Boundary Terms
    3. More on Fourier Transforms
    4. Applications to Wire Antennas
      1. On the Kernels of Hallén's and Pocklington's Equations
      2. Behavior of Current Near Delta-Function Generator
    5. Problems
    6. References
  7. Poisson Summation Formula and Applications
    1. Doubly Infinite Sums
      1. Formula and its Derivation
      2. Remarks
      3. A First Example
      4. Application: Infinite Linear Array of Traveling-Wave Currents
      5. Application: Coupled Pseudopotential Arrays
    2. Finite Sums
      1. Formula and Proof
      2. Remarks
      3. Elementary Example
      4. Continuous Functions with Equal Endpoint Values
      5. Application: Cylindrical Array of Traveling-Wave Currents
    3. Problems
    4. References
  8. Mellin-Transform Method for Asymptotic Evaluation of Integrals
    1. Summary of Mellin-Transform Method
    2. Lemmas for Residue Calculations
    3. Simple Example
    4. On the Convergence of Mellin-Barnes Integrals
    5. Application to Highly Directive Current Distributions
    6. Further Reading
    7. Problems
    8. References
  9. More Applications to Wire Antennas
    1. Problem Pertaining to Magnetic Frill Generator
      1. Statement of Problem
      2. Preliminaries
      3. Derivation of Eq. 8.6
    2. Oscillations with the Approximate Kernel: Case of Delta-Function Generator
      1. Integral Equation: Nonsolvability
      2. Numerical Method: Solution for Nonzero Discretization Length
      3. Asymptotic Approximation for Small Discretization Length
    3. On the Near Field Due to Oscillating Current
      1. Statement of Problem
      2. Derivation of Eq. 8.46
    4. Supplementary Remarks
    5. Problems
    6. References
  10. Special Functions
    1. Preliminaries
    2. Exponential, Sine, and Cosine Integrals
      1. Definitions and Small-Argument Expansions
      2. Large-Argument Expansions
    3. Complete Elliptic Integral of the First Kind
    4. Bessel and Hankel Functions
      1. Definitions and Small-Argument Asymptotic Approximations
      2. Large-Argument Asymptotic Expansions
      3. Large-Order Asymptotic Approximations
      4. Addition Theorem for Hankel Function of Order Zero
    5. Modified Bessel Functions
    6. Generalized Hypergeometric Functions
    7. Problems
    8. References
  11. On the Convergence/Divergence of Definite Integrals
    1. Some Remarks on Our Rules
    2. Rules for Determining Convergence/Divergence
    3. Examples
    4. References
  12. Authors' Biographies
  13. Index (1/2)
  14. Index (2/2)

Product information

  • Title: Selected Asymptotic Methods with Applications to Electromagnetics and Antennas
  • Author(s): George Fikioris, Ioannis Tastsoglou, Odysseas N. Bakas
  • Release date: September 2013
  • Publisher(s): Morgan & Claypool Publishers
  • ISBN: 9781627050401