VII—Further Investigations into the Periodic Lamé Functions

Hitherto the Lamé functions of u , of real periods 2 K and 4 K , and the Lamé polynomials in particular, have been expressed as series of integral powers of sn u (multiplied, in appropriate cases, by cn u , dn u , or cn u dn u ). The necessity to use these, or the corresponding Weierstrassian forms, is a serious handicap to research, because the terms of the expression are not mutually orthogonal. In particular, results that can be deduced from the orthogonal properties of the Fourier series in which periodic Mathieu functions are developed, find no analogues in the Lamé functions as expressed in the customary forms. For instance, the integral equations that generate the Mathieu functions have proved to be valuable sources of information, but those for the Lamé functions cannot be much more than scientific curiosities until the development of Lamé functions as series of orthogonal functions enables them to be put to use.

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