B.S 1964, ch.20) in analogy with the relation between the Bessel and the modified Bessel equations The

designation Mathieu functions is generally reserved for those solutions of (5b) which have periodicity If or

XIT

X , termed eigenvalues, which depend on k and on the order of the functions.

8.2 Brief survey of earlier work» and applications of Mathieu*s equation to problems in physics

and engineering;.

Mathieu*s equation and its solutions were first studied by Mathieu (1868) in connexion with the deter­ mination of the natural frequencies for a stretched membrane of elliptical shape. As the eccentricity of the elliptic boundary tends to zero, the problem becomes essentially that of solving Bessel*s equation. It is thus to be expected that there exists a close

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relationship between Mathieu and Bessel functions. In fact, Heine (1878) showed that one set of periodic sol­ utions of (8.1.5b) could be expanded in a series of Bes­ sel functions. Since those early days, iviathieu's equat­ ion has found application in a diversity of mathematical and physical problems.

From the mathematical point of view, the Mathieu equation is of considerable importance as the simplest representative of a host of differential equations with

one or more periodic coefficients. These include, for

instance, prolate and oblate spheroidal wave equations

and Lame*s equation. To quote Professor Brdelyi in

his foreword to the National Bureau of Standards tables (1951) : '* Among all the special functions arising

out of the separation of the partial differential equat­ ions of mathematical physics in various systems of co­ ordinates, Mathieu functions are the first to lead def­ initely outside the circle of hypergeometric and allied functions; hence the difficulties encountered in their theory and nuraerical computation alike, and hence the necessity of using new methods.”

•175-

equation arises essentially in one of two ways: i) separation of the wave equation for a three-dimensional problem with elliptic cylindrical symmetry , or for a two-dimensional problem with elliptical, symmetry; and

ii) study of one-dimensional problems involv­ ing periodically varying parameter(s). The following

may be cited as examples of the first category: diffract­ ion of electromagnetic waves by an elliptical cylinder

(Sieger 1908), electromagnetic waves in metal cylinders of elliptic cross-section (Chu 1938), elliptic antennae

(Schelkunoff 1952), free oscillations of water in ellipt­ ical lakes (Jeffreys 1924), stability of columns and

strings under periodically varying forces (Lubkin and Stoker 1943), aerofoil in a windtunnel of elliptic section (Rosenhead 1933), motion of elliptic cylinders through viscous liquid (Harrison 1924), vibration of elliptic cylinder in viscous fluid (Ray 1936), steady flow of viscous fluid past an elliptic cylinder (Tomot- ika and Aoi 1953), skin effect in cylindrical conduct-

+

ors (Strutt 1928), wave mechanical theory of the

ion (Teller 1930, Hylleraas 1931, Johnson 1941), thermo­ dynamic functions for molecules having restricted

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internal rotations (Pitzer 1937, Li and Pitzer 1956), restricted rotations of molecules and tunnelling through periodic barriers (Das 1957, Stejskal and Gutowski 1958). Some problems belonging to the second class are; stab­ ility of periodic motion (Thomson 1892a,b), maintenance of vibrations (Raman 1912), frequency modulation (Carson 1922,. Barrow 1934, Brdelyi 1934, Barrow, Smith and Bau­ mann 1936), stability of non-linear oscillators (McLach- lan 1951), the physical pendulum in wave mechanics (Con­ don 1928).

Several treatises on the theory and application of Mathieu functions have been published, The monograph by Strutt (1932) is useful for its extensive biblio­ graphy of papers published prior to 1932. In his book on the theory and application of Mathieu functions,

McLachlan (1947) has sacrificed some mathematical rigour to produce a reference work which covers in useful det­ ail the applied mathematics, physics and engineering

aspects of Mathieu functions. An extensive bibliography is given in addition to many new results. The book by Meixner and Schafke (1954): *Mathieusche Funktionen und

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analytical standpoint. There are also long chapters on Mathieu functions in volume III of the Bateman manu­ script * Higher transcendental functions * prepared by Erdalyi and others (1955) * In addition, i'lathieu funct­ ions are studied extensively in such texts as; *A course in modern analysis* by Whittaker and Watson (1927) and

'Methods of theoretical physics' by Morse and Feshbach

(1953). Some early tables on Mathieu functions, and

their characteristic numbers, were published by Goldstein (1927), but the most extensive tables available for a long time were those of Ince (1932), which also include

values of the zeros and turning points. More recent

tables giving Fourier coefficients of Mathieu functions and their characteristic numbers are those of Stratton, Morse, Ghu and Hutner (1941), and the National Bureau

of Standards (1951; these tables also give joining fact­ ors relating to the various solutions of Mathieu*s equat­

ion). In the Introduction to the latter work, Dr.G.Blanch

gives an excellent account of the more practical aspects of the theory of Mathieu functions.

8.3 The inte^ro-differential equation.- Let the Mathieu equation be written

«176-

Èbk-.

where is large and positive and X represents the characteristic numbers. The form of the asymptot- ic expansion for

\

Dingle and Muller 1962), and for later convenience we write

oo

‘

X

zlv 4r ^ --- > (2)

* j " (-‘b y

where w> is approximately an odd integer, and ol «

«