Graphs and Tables of the Mathieu Functions and Their First Derivatives

Mathieu functions are employed in solving a variety of problems in mathematic (al?) physics. In many cases the configuration involves elliptical coordinates. Of course, the circular geometry is the degenerate case of the elliptical cross section. 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 author's 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 I 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 Emile 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.