Proceedings of the Edinburgh Mathematical Society

Table of Contents - Volume 41

Research Article

Associated Mathieu Functions

Dr E. L. Ince

The periodic solutions of the linear differential equation

S001309150000362X_eqnU1

,

which reduce to Mathieu functions when v = 0 or 1, will be known as the associated Mathieu functions. The significance of this terminology will appear in the following section.

(Received February 17 1922)

(Accepted May 11 1922)

Notes

* Math. Annalen, 52 (1899) pp. 81–112. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

† Proc. R.S.E., 42 (1922) p. 47. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

‡ Proceedings, 40 (1922) pp. 28–29.

* I.e. a regular singularity with exponent-difference ½. It is to be remembered that the coalescence of two elementary singularities produces in general a regular singularity with arbitrary exponent-difference; the coalescence of three elementary singularities generates an irregular singularity of the first speoies, and so on.

† When v = ½ the equation bears the same relation to Legendre's equation as its general form bears to the associated Legendre equation.

‡ It has bean proved by the present writer, Proc. Camb. Phil. Soc., 21 (1922) p. 117 [OpenURL Query Data]  [Google Scholar], that when v=0 or 1 the equation cannot have two periodic solutions except for θ=0. It is shown in the present section that this is true for all values of v.

* Hermite, Crelle's Journal, 89 (1891) p. 18 [OpenURL Query Data]  [Google Scholar], Œuvres, 4 p. 18. A still further generalisation of Lamp's equation is given by Darboux, Comptes rendus 1882, and de Sparre, Acta Math. 4, (1883) pp. 105–140, 289–321. [OpenURL Query Data]  [Google Scholar]