Accurate Computation of Mathieu Functions

Book description

This lecture presents a modern approach for the computation of Mathieu functions. These functions find application in boundary value analysis such as electromagnetic scattering from elliptic cylinders and flat strips, as well as the analogous acoustic and optical problems, and many other applications in science and engineering. The authors review the traditional approach used for these functions, show its limitations, and provide an alternative "tuned" approach enabling improved accuracy and convergence. The performance of this approach is investigated for a wide range of parameters and machine precision. Examples from electromagnetic scattering are provided for illustration and to show the convergence of the typical series that employ Mathieu functions for boundary value analysis.

Table of contents

  1. Introduction
  2. Mathieu Functions
    1. The Mathieu Equations
    2. Angular Functions
      1. Relations Satisfied by the Expansion Functions { A } and { B }
    3. Radial Functions
    4. Computational Steps
    5. Summary
  3. Observed Accuracy Using Traditional and Tuned Methods
    1. Angular Functions
      1. Subtraction Error
      2. Back Substitution
    2. Radial Functions (1/2)
    3. Radial Functions (2/2)
    4. Example: Computing a Hankel Function in Terms of a Summation of Mathieu Functions
    5. Summary
  4. Recommended Algorithm for Mathieu Function Computation
    1. The Tuned Algorithm
    2. Example: Calculation of a Uniform Plane Wave
    3. Adaptive Error Estimation Based on the Plane Wave
    4. Summary
  5. Electromagnetic Scattering from Conducting Elliptic Cylinders
    1. The TM_z Case
    2. The TE_z Case
    3. Examples
    4. The Size, N, of the Eigenmatrix for the Computations to Follow
    5. Current Density and Scattering Cross Section for TM_z Excitation (1/3)
    6. Current Density and Scattering Cross Section for TM_z Excitation (2/3)
    7. Current Density and Scattering Cross Section for TM_z Excitation (3/3)
    8. Current Density and Scattering Cross Section for TE_z Excitation (1/3)
    9. Current Density and Scattering Cross Section for TE_z Excitation (2/3)
    10. Current Density and Scattering Cross Section for TE_z Excitation (3/3)
    11. Summary
  6. Electromagnetic Scattering from an Infinite Conducting Strip
    1. The TM_z Case
    2. The TE_z Case
    3. Results (1/3)
    4. Results (2/3)
    5. Results (3/3)
    6. Summary
  7. Converting Between Two Common Mathieu Function Conventions
  8. Tables of Select Eigenvalues and Eigenvectors (1/2)
  9. Tables of Select Eigenvalues and Eigenvectors (2/2)
  10. Tables of Select Angular Mathieu Functions (1/2)
  11. Tables of Select Angular Mathieu Functions (2/2)
  12. Tables of Select Radial Mathieu Functions (1/2)
  13. Tables of Select Radial Mathieu Functions (2/2)
  14. References
  15. Authors' Biographies

Product information

  • Title: Accurate Computation of Mathieu Functions
  • Author(s): Andrew Peterson, Malcolm Bibby
  • Release date: September 2013
  • Publisher(s): Morgan & Claypool Publishers
  • ISBN: 9781627050869