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Associated Mathieu Functions

Published online by Cambridge University Press:  20 January 2009

E. L. Ince
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The periodic solutions of the linear differential equation

,

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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , February 1922 , pp. 94 - 99
Copyright
Copyright © Edinburgh Mathematical Society 1922

References

* Math. Annalen, 52 (1899) pp. 81–112.CrossRefGoogle Scholar

† Proc. R.S.E., 42 (1922) p. 47.Google Scholar

‡ Proceedings, 40 (1922) pp. 28–29.Google Scholar

* 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 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 Google Scholar, Œuvres, 4 p. 18.Google Scholar 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.Google Scholar