By Jean-Pierre Aubin
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Additional info for Approximation of Elliptic Boundary-value Problems
NS N(e Z) ]res f(Zi) dz } CHAPTE RIEMANN SPHERE The symbol ao is next introducedt into complex analyBy analogy with real analysis we would set 1/0 = oo, but in complex analysis, + co and - co must have the same sis. 5 0) must also mean the same thing. ) Yet the algebraic convenience of a symbol for 1/0 is not enough justification. ) For the complete reason we must also turn to geometry. TREATMENT OF INFINITY 2-1 Ideal Point Let us consider the fact that the z plane is not compact. This means that certain sequences z1, z2, .
Here the difficulty is that the right- and left-hand sides of e are connected on completing a circuit of 0. We shall ultimately see that this situation can be completely avoided in complex variable theory. TORUS Consider the surface of revolution of, say, the circle (t - a)2 + l'2 = b2 (1-25) about the i' axis in (real) ( e: azimuthal circles e*: longitudinal circles space. (on a>b>0 Then, these loci = const) = const) on 77 constitute simple closed curves which still do not serve as boundaries of any region.
3 Let us refer to Fig. 1-3a again. Show how to draw a closed curve e -' mle, + mze2 for arbitrary integers m,, ms. 4 For an n-tuply-connected region 61, show that a closed curve is generally + mn-1C,i_, by generalizing the diagram in Fig. 1-3a. homologous to mlel + SINGULAR BEHAVIOR Cauchy Kernel 1-7 The completion of the proof of Theorem 1-1 is based on the following result : THEOREM 1-6 (Cauchy's representation theorem) If a simple closed curve e, positively oriented and lying in the region 61, contains only points z of 61 in its interior, then a function f (z) analytic in CR can be represented for points z interior to e by f(z) = 1 r z dZ (1-33) taxfe Z - z As we recall, any operation we desire to perform on f (z) becomes transferred through the integral sign to the so-called "Cauchy kernel" C(Z,z) = Z/2xz (1-34) Formally, at least, we find that by taking the nth derivative, we obtain (1-35) d dz) _ 1 Z - z) n!
Approximation of Elliptic Boundary-value Problems by Jean-Pierre Aubin