James C. Wiltse - Graphs And Tables Of The Mathieu Functions And Their First Derivatives

GRAPHS AND TABLES OF THE
MATHIEU FUNCTIONS
AND THEIR FIRST DERIVATIVES

BY

JAMBS C. WILTSE

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First published by AuthorHouse 01/16/2012

ISBN: 978-1-4685-4462-6 (sc)

ISBN: 978-1-4685-4461-9 (ebk)

Library of Congress Control Number: 2012900919

Printed in the United States of America

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Graphs and Tables of the

Mathieu Functions

and their First Derivatives

James C. Wiltse

Acknowledgement

Dr. Marcia J. King made significant contributions to this document. She calculated many of the tabulated values and verified the analysis for Section I and II. Her assistance was much appreciated.

The author would like to acknowledge the editorial assistance of Dr. Helen C. Wiltse in preparing the material for this volume.

Contents

Mathieu functions are employed in solving a variety of problems in mathematical physics. In many cases the configuration involves elliptical coordinates. Of course, the circular geometry is the degenerate case of the elliptical cross section.

A typical example is that of cylindrical waveguides for electromagnetic waves. This includes hollow metal guides, coaxial transmission lines, and surface waveguides (a dielectric rod or a metal wire) of elliptical cross section. Solutions of the wave equation involve products of angular periodic and radial Mathieu functions. Similarly, circular waveguides involve products of sinusoids and Bessel functions, which the Mathieu functions smoothly transition to as the ellipse becomes a circle.

This volume contains values for, and curves of the angular and radial Mathieu functions and their first derivatives. The latter are often needed in the solution of problems, in particular in solving electromagnetic wave propagation problems. Also included are data on zero crossings of the radial Mathieu functions. These are often needed for determining the cut-off frequencies for propagating modes.

Other tables are available for the Mathieu functions, but there is very little data available for derivatives or zero crossings. It is felt that the principal value of this volume is in the multitude of curves included. The analyst dealing with elliptical cases can, by inspection of the curves, find values of the functions and derivatives at the origin, maxima and minima, zero crossings, and qualitative behavior of the plots as a function of several parameters. To the authors knowledge, this is the most extensive presentation of plotted information. It is hoped that the information will be helpful in the solution of practical problems.

This book is divided into two sections. Section 1 deals only with the functions themselves, defining the equations and terminology used and presenting the tabular data and curves. Section II treats the derivatives and the zeros. Again the terminology and equations for the first derivatives are given.

The Mathieu functions are named after Emiie L. Mathieu (1835-1890), a French mathematician, who in 1868 published an article dealing with vibratory movement of the elliptic membrane. The asteroid 27947 Emilemathieu is named in his honor.

Summary

This section contains tabulated values and curves for some of the Mathieu functions. Included are the periodic Mathieu functions Sen(s, v) and Son(s, v) for integer orders 0 < n < 2 and various positive and negative real values of the parameter s. Also, included are the radial Mathieu functions Jen (s,u), Jon (s,u), Nen (s,u), and Non (s,u)with s positive real, and Hen (1) (s,u) and Hon (1) (s,u) with s negative real, for 0 < n (integer) < 2 and 0 < u < 2 in each case.

Section I
Perlodic Angular and Radial Mathleu Functions

LIST Of
ILLUSTRATIONS

Figure 1 The Elliptic Cylinder Coordinate System, The Positive Direction is out of the Page

2 Even Periodic Mathieu Function of Order Zero

3 Even Periodic Mathieu Function of Order One

4 Even Periodic Mathieu Function ofOrder One

5 Even Periodic Mathieu Function ofOrder One

6 Even Periodic Mathieu Function ofOrder Two

7 Even Periodic Mathieu Function ofOrder Two

8 Odd Periodic Mathieu Function ofOrder One

9 Odd Periodic Mathieu Function ofOrder One

10 Odd Periodic Mathieu Function ofOrder Two

11 Odd Periodic Mathieu Function of O rder Two

12 Even Periodic Mathieu Function ofOrder Zero and Negative Parameter s

13 Even Periodic Mathieu Function ofOrder One and Negative Parameter

14 Even Periodic Mathieu Function ofOrder One and Negative Parameter s

15 Odd Periodic Mathieu Function ofOrder One and Negative Parameter s

16 Even Periodic Mathieu Function of Order Two with Negative Parameter s

17 Odd Periodic Mathieu Function of Order Two with Negative Parameter s

18 Even Radial Mathieu Function of the First Kind and Order Zero

Kind and Order Zero *

19 Even Radial Mathieu Function of the First Kind and Order Zero

20 Even Radial Mathieu Function of the First Kind and Order One

21 EvenRadial Mathieu Function of the First Kind and Order One

22 EvehRadial Mathieu Function of the First Kind and Order Two

23 Even Radial Mathieu Function of the First Kind and Order Two

24 Odd Radial Mathieu Function of the First Kind and Order One

25 Odd Radial Mathieu Function of the First Kind and Order One

26 Odd Radial Mathieu Function of the First Kind and Order Two

27 Odd Radial Mathieu Function of the First Kind and Order Two

28 Even Radial Mathieu Function of the Second Kind and Order Zero

29 Even Radial Mathieu Function of the Second Kind and Order Zero

30 EvenRadial Mathieu Function of the Second Kind and Order One

31 Even Radial Mathieu Function of the Second Kind and Order One

32 Even Radial Mathieu Function of the Second Kind and Order Two

33 Even Radial Mathieu Function of the Second Kind and Order Two

34 Odd Radial Mathieu Function of the Second Kind and Order One

35 Odd Radial Mathieu Function of the Second Kind and Order One

36 Odd Radial Mathieu Functions of the Second Kind and Order Two

37 Odd Radial Mathieu Function of the Second Kind and Order Two

18 Even Radial Mathieu Function of the Third Kind and Order Zero, for Negative Real Values of the Parameter s

19 Even Radial Mathieu Function of the Third Kind and Order One, for Negative Real Values of the Parameter s

20 Even Radial Mathieu Function of the Third Kind and Order Two, for Negative Real Values of the Parameter s

21 Odd radial Mathieu function of the Third Kind and Order One, for Negative Real Values of the Parameter

22 Odd Radial Mathieu Function of the Third Kind and Order Two, for Negative Real Values of the Parameter s