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For further analysis we define an angular parameter when A is real; the arguments are all to be taken as 0 when A = 0, and calculated thence by continuousâ€™ variation. ) =a + 2rv. 5) Here r will run through some sequence A of consecutive integers; in the event that 03 is an eigenvalue, we may for definiteness take as representative, and incorporate the corresponding r-value in A . T h e eigenvalues thus defined are, of course, dependent on a and on m, +-= A,. 6) and an important group of topics centers on the dependence of h, on m and a, and more generally on the dependence on m and a of the spectral function, to be defined shortly.

BOUNDARY PROBLEMS FOR RATIONAL FUNCTIONS sional cases by way of a limiting process. 6. 2-3) be all finite. 7-8) follows standard lines, and is given in Appendix 111. 12), we have the dual expansion theorem, likewise purely elementary. 7. 2-3) have only finite eigenvalues. 15). It is a simple matter to deduce one of these expansion theorems from the other; the situation naturally becomes much more difficult in infinitedimensional cases, even when both expansions are discrete. 6) holds. 7-8). 5. 19) where X, is any of the finite eigenvalues.

We pursue this example to the extent of setting up an inhomogeneous problem, leading to the characteristic function. We suppose a force F cos ( w t ) applied transversely to the particle a, , the string being pinned down at both ends (Fig. 1). Assuming, if possible, the displace~ F C OwSt FIG. 1 . Inhomogeneous problem for vibrating string. 8) = 0, and finally v, = 0. 6). 8) if we arrange that + F = C-10-1 = c-l(uy-1 + /%-I) = Pc-1 , 18 0. INTRODUCTION by the initial conditions. We deduce that + /lzm= 0, we The function -z,(h)/ym(h) will be a special case of what we use in Chapters 4 and 5 as a characteristic function; it will be expedient to use a more general function, corresponding to a more general boundary condition at the upper end, with a view to limit-circle investigations in the complex plane.