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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

REFERENCES TO LITERATURE

1.Besicovitch, A. S., 1932. Almost-periodic functions. Cambridge Univ. Press.Google Scholar
2.Elliott, W. W., 1928, 1929. “Green's functions for differential systems containing a parameter”, Amer. Journ. Math., L (1928), 243–258; Li (1929), 397–416.CrossRefGoogle Scholar
3.Erdélyi, A., 1941. “On Lame functions”, Phil. Mag., xxxi, 123–130.CrossRefGoogle Scholar
4.Ince, E. L., 1925–1927. “Researches into the characteristic numbers of the Mathieu Equation”. Part I, Proc. Roy. Soc. Edin., XLVI (1925), 20–29. Part II, Proc. Roy. Soc. Edin., XLVI (1926), 316–322. Part III, Proc. Roy. Soc. Edin., XLVII (1927), 294–301.Google Scholar
5.Ince, E. L., 1927. “The Mathieu equation with numerically large parameters”, Journ. London Math. Soc., II, 46–50.CrossRefGoogle Scholar
6.Ince, E. L., 1940. “The periodic Lamé functions”, Proc. Roy. Soc. Edin., LX, 47–63.CrossRefGoogle Scholar
7.Ince, E. L., 1940. “Further investigations into the periodic Lamé functions, Proc. Roy. Soc. Edin., LX, 83–89.CrossRefGoogle Scholar
8.Langer, R. E., 1934. “The asymptotic solutions of certain linear ordinary differential equations of the second order”, Trans. Amer. Math. Soc., xxxvi, 90–106.CrossRefGoogle Scholar
9.Langer, R. E., 1934. “The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to the Stokes' phenomenon”, Bull. Math. Soc., XL, 545–582.CrossRefGoogle Scholar
10.Langer, R. E., 1935. “On the asymptotic solution of ordinary differential equations, with reference to the Stokes's phenomenon about a singular point”, Trans. Amer. Math. Soc., XXXVII, 397–416.Google Scholar
11.Mulholland, H. P., and Goldstein, S., 1929. “The characteristic numbers of the Mathieu equation with purely imaginary parameter”, Phil. Mag., VIII, 834–840.CrossRefGoogle Scholar
12.Poincaré, H., 1895. Analytic theory of the propagation of heat (French), Paris.Google Scholar
13.Strutt, M. J. O., 1943. “Bounds for the characteristic parameter-values corresponding to problems of Hill. Part I. Characteristic values of smallest moduli”, Proc. Roy. Acad. Amsterdam, LIl, 83–90.Google Scholar
14.Strutt, M. J. O., 1943. Idem. Part II. “Characteristic values of any order” (in Dutch), Proc. Roy. Acad. Amsterdam, LII, 97–104.Google Scholar
15.Strutt, M. J. O., 1943. “Curves of characteristic parameter-values corresponding to problems of Hill. Part I. General character of the curves” (in Dutch), Proc. Roy. Acad. Amsterdam, LII, 153–162.Google Scholar
16.Strutt, M. J. O., 1943. Idem. Part II. “Asymptotic character of thecurves” (in Dutch), Proc. Roy. Acad. Amsterdam, LII, 212–222.Google Scholar
17.Strutt, M. J. O., 1943. “Characteristic functions corresponding to problems of Hill. Part I. Completeness of the sets of periodic and of almost-periodic characteristic functions” (in Dutch), Proc. Roy. Acad. Amsterdam, LII, 488–496.Google Scholar
18.Strutt, M. J. O., 1943. Idem. Part II. “Expansion formulas in series of periodic and of almost-periodic characteristic functions” (in Dutch), Proc. Roy. Acad. Amsterdam, LII, 584–591.Google Scholar
19.Strutt, M. J. O., 1932. Lamé's, Mathieu's and related functions in physics and engineering (booklet), Springer, Berlin.Google Scholar
20.Strutt, M. J. O., 1934. “Hill's differential equation in the complex domain” (in German), Nieuw Archief voor Wiskunde, XVIII, 31–55. Abstract in Comptes Rendus Paris Acad., CXCVIII, 1008-1010.Google Scholar
21.Tamarkine, J., 1912. “On some points of the theory of ordinary linear differential equations and on the generalization of Fourier's series” (French), Rend, del Circolo Mat. di Palermo,XXXIV, 345–382.CrossRefGoogle Scholar
22.Whittaker, E. T., and Watson, G. N., 1940. A course of modern analysis, Cambridge.Google Scholar
23.Strutt, M. J. O., 1944. “Real eigen-values of Hill's problems of the second order” (German), Math. Zeits., XLIX, 593–643.Google Scholar