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XXX.—On Hill's Problems with Complex Parameters and a Real Periodic Function

Published online by Cambridge University Press:  14 February 2012

M. J. O. Strutt
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Summary

Hill's differential equation (1.1) derives its importance from being the prototype of the different equations of Lamé and of the equation of Mathieu, which are connected with wave and potential problems in mathematical physics. Besides this, numerous instances of its occurrence in problems of elasticity and of dynamical or statical stability are known. In the present treatment, conditions are reversed with respect to most of the older publications, since the characteristic multiplier σ of equations (1.2) is not sought as a function of the given parameters λ and γ of equation (1.1), but σ is supposed given and the corresponding values of λ and γ are regarded as unknown. Thus a linear homogeneous boundary value problem of the second order and of non-self-adjoint type ensues, the values of σ and of λ, σ being in general complex. On this latter point the present paper considerably enlarges the scope of some previous papers published by the author during the war along somewhat similar lines but for real characteristic values (Nos. 13–18 of the references at the end).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 3 , 1948 , pp. 278 - 296
Copyright
Copyright © Royal Society of Edinburgh 1946

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References

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