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By Garrett Birkhoff

A learn of the artwork and technology of fixing elliptic difficulties numerically, with an emphasis on difficulties that experience very important clinical and engineering purposes, and which are solvable at reasonable expense on computing machines.

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It is easily shown that any two eigenfunctions having distinct eigenvalues are orthogonal both with respect to P ( w , w ) and with respect to Q{ « , « ) . ) Sommerfeld has named this assertion, which was first proved in appropriate generality after 1900 by Hilbert, Weyl, and others, the 'Ohm-Rayleigh principle'. 15 15 See A. Sommerfeld, 179. Partial Differential Equations in Physics, Academic Press, 1949, p. CLASSICAL ANALYSIS 41 7. Maximum principle. We now turn our attention to general linear elliptic differential operators with variable coefficients, having continuous coefficient-functions in a compact domain O.

171]). THEOREM 12. Let w = / ( z ) be an analytic function of the complex variable 2 in a domain fl of the upper half-plane, whose boundary F includes a segment S of the real axis. Let w be continuous in H and real on S. Then f can be continued analytically into the mirror image ft' of O in the lower half-plane by setting w = [/(z*)]* there, where z* designates the complex conjugate of z. COROLLARY 1. Let u ( x , y ) € C ( f l ) be harmonic in ft, continuous in ft, and let w ( x , 0 ) = 0 on S.

P. Eisenhart, Annals of Math. , Symmetry and Separation of Variables, Addison-Wesley, 1977. CLASSICAL ANALYSIS 25 solve the Poisson equation in B for general Dirichlet-type boundary conditions. 2. Complex variable techniques. In §1, we described some elliptic problems which are solvable in terms of tabulated functions of one real variable. , functions satisfying uxx + uyy = Q), are most easily solved in terms of functions of one complex variable. This is because the real and imaginary parts of any complex analytic function w = /(z) of a complex variable z=x+iy are conjugate harmonic functions of the two real variables x and y.

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