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Introduction. The purpose of this paper is to describe an effective construction of Greene and Neumann's functions for a general class of linear, second-order partial differential equations of elliptic type in terms of a set of continuously differentiable functions, complete and orthonormal with respect to the domain considered. This result completes a previous paper 2 by us in which an analogous construction was given for the difference between the Green and Neumann functions. In order to obtain representations of the Green and Neumann functions separately, one had to assume in the former paper the knowledge of a fundamental solution of the differential equation considered. Since it is by no means an easy task to construct for a given linear partial differential equation of second order a fundamental solution, and since the method of this paper permits a direct construction of the Green and Neumann functions without knowledge of such a fundamental solution, our result seems to constitute a certain advance in the application of orthonormal sets to the theory of linear partial differential equations. Generalities and notations.We choose a finite domain B in the plane (x, y) which is bounded by a set of M closed smooth curves For the sake of a concise notation, we shall throughout this paper denote a point (x, y) of the domain B by Z and the corresponding complex number x+iy by z. Another point (u, v) of B will be denoted by TV and analogously u+iv = w; in the same way, (r, s) = T, r+is = t.A function £(Z, W) is called a fundamental solution of (1) in B if it is a solution of (1) as a function of Z£J3 for fixed W£J3, which is finite everywhere in B with the exception of the point W, where it has a logarithmic singularity. It may be written in the form (3) 5(Z, W) = A(Z 9 W) log -: r + a(Z t W)
1. The potential and stream functions, 4 and 4,', of an irrotational steady flow of a compressible perfect fluid satisfy a system of non-linear equationswhere p is the density. (See [2.1].1) Under the usual hypotheses p = p(OX2+O)2, k) is a function of 4)z2 + q,2 which depends upon a parameter k. This parameter enters in the equation of state, which we assume to have the form p = a + cpk, a, c and k being constants. Let [v exp (i@)] denote the velocity vector. If we introduce v and 9 as independent variables then the equations connecting 4 and 4, become linear. As Chaplygin showed the above system becomes v1lp4e -4', = 0, p-tv-1(l -M2)4,0 + 0, = 0. [3.8],(1.1) Here the Mach function, M(v, k), and the density, p(v, k), are given functions of v. Eliminating 4) from (1.1) yields S(4,) p-2(1 -M2)4,'0 + p-lv[p-14,],. = 0. [2.6],(1.2) In the case of an incompressible fluid, 4' is a harmonic function of 0 and log v. Taking the imaginary part of an analytic function of the complex variable (log v -i@) one obtains the stream function of a possible incompressible fluid flow. In § 2 and § 4 we shall define operators, P, which transform analytic functions of one complex variable into solutions 4'(v, 0) of S(4') = 0. Thus we obtain a method for generating possible stream functions of compressible fluid flows. The computing of the corresponding stream function in the physical plane, i.e., of 4[v(x, y), 0(x, y)], is reduced to quadratures. (See § 5 of [1].) We note, however, that the obtained flows do not always have a physical significance. (See § 4 of [1].)2. Chaplygin was the first to introduce operators into compressible fluid theory. Let 2B.7r' exp [i(2n0 + a,)], r= 1/2(k -1)ao-Iv2, B,, 2 0, be the series development of an analytic function. Then the relation 4 + i4y = ZBxeTY,oYxj-(1 + n-l1rYx'Yx-l)(1 -T)Pcos (2n0 + a,,) + i sin (2n + a,,)] (2.1) 276 PROC. N. A. S.
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