By Robert B. Burckel

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It follows from (9, (8) and (9) that K c K(n, m). 43 Let U be an open subset of C and C1,. , C, distinct bounded components of C\U. Show that there exists compact K c U such that C,, . , C, lie in distinct bounded components of C\K. Hints: Choose bounded, open V, 2 C, such that Vl, . , V, are pairwise disjoint. 34 to select compact K, c C\V, relatively open in C \ V and satisfying C, c K, c V,. C\(U U Kl U * U KM)= (C\U)\(Kl U U K,) is -- - 6. Extension Theorems 37 relatively closed in @\U (since the K, are relatively open there) and so is closed in C.

This is the work of the next two chapters. In FOUBT [1904] the reader will find a detailed comparison of the three theories with many references to the original works and in PRINGSHEIM [1925], [1932] a complete and careful (though sometimes pedantic) presentation of the Weierstrass theory. ) For [I9751 and for the the historical evolution of Weierstrass’ theory see MANNING [ 19251. master’s own comparison of the three approaches, WEIERSTRASS 53 Chapter I11 Power Series and the Exponential Function $1 Introduction As with integrals, power series (though fascinating) are a tool here and are not pursued extensively.

L), is connected. Define (4) K(n, rn) = C\U(n, m). Thus K(n, m) is a compact set which is disjoint from B(0, 1) and has connected complement. Now let K be any simply-connected, compact subset of C\B(O, 1). Choose r > 1 + suplKI. Then 1 and r are two points in the open connected set C\K. 28 there is a continuous y : [0, 11 + C\K with y(0) = 1, r(1) = r . Choose a positive integer m so large that D(0, 1 (5) + l/m) c C\K, and (6) 2 ; < dist(A0, 11, K ) . 11 and also that < Ifn(1)l IIfn(l)l + l/m - 1 - l/m (if m 2 21, that is, (9) K = D(0, Ifn(1)l - 1h).