Advanced Calculus fi..

700 Advanced Calculus, Fifth Edition-inside R and have certain new boundary values on the boundary of R. Determinationof v is then a Dirichlet problem, which can be attacked by conformal mapping(Section 8.22). While conformal mapping is not available as a tool for the correspondingproblems in three dimensions, methods based on potential theory canbe used; see especially the book of Kellogg listed at the end of the chapter. (Anintroduction to potential theory is given in Sections 5.16 to 5.18.)One can in general reduce the equilibrium problem v2u = - F to the case ofzero boundary values by the following procedure (cf. Section 10.12): Let u(x, y) berequired to have values h(x, y) on the boundary C of R. If h(x, y) is sufficientlysmooth, one can then find a function h (x, y) that has continuous first and secondderivatives inside R, is continuous in R plus C, and equals h(x, y) on C. Thefunction v = u - hl is then zero on C, and V2v = - F - V2hl = - F1(x, y). Thedetermination of v can be carried out with the aid of a Green S function, as will beindicated in Section 10.18. (See also Sections 5.17 and 5.18.)In many physical problems it is natural to regard the continuous medium as beinginjnite in extent. For example, in one dimension, one can consider the wave equationfor the infinite interval x > 0. If one seeks normal modes, one is led to the characteristicvalue problem:This problem has solutions for every value of A, namely, the functions sinax, fora2a2 = h2. Thus the resonant frequencies A form a "continuous" set of numbers,and one has a "continuous spectrum." [There also exist unbounded "normal modes":u = sinh a.3: eUa'. These are of less physical interest.]One can construct linear combinations of the normal modes in order to obtaina "general solution" of the homogeneous problem. Since there is a continuous sequenceof A's, an integration is called for rather than a summation. For a 1 0 wemust integrate expressions of formsin x[p(a) cos (aat) + q(a) sin (aat)],where p(a) and q(a) are "arbitrary" functions of a. We obtain the integrallrn sinux[p(a) cas (aar) + q(a) sin (oar)] da.For each fixed t this can be considered as a Fourier integral (the Fourier sine integral,Section 7.18). In particular, for t = 0 we obtain a Fourier integral representation ofthe initial displacement u(x, 0):lrn p(a) sin ax do. - ..