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Associated Mathieu Functions

Published online by Cambridge University Press:  20 January 2009

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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
Copyright
Copyright © Edinburgh Mathematical Society 1922

References

* Math. Annalen, 52 (1899) pp. 81112.CrossRefGoogle Scholar

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

Proceedings, 40 (1922) pp. 2829.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. 105140, 289321.Google Scholar