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By Robert B. Burckel

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Power Series and the Exponential Function 54 (ii) (iii) The terms of the series (1) are unbounded (and hence the series diverges) for every z E C with Iz - zol > R. The extended real number R is the reciprocal of the extended real number and is called the radius of convergence of the series (1). limndm Proof: Define R to be the least upper bound in [0, 031 of the r E [0, 00) for which {Icnlr"} is a bounded sequence. If Iz - zol > R, then the sequence {c,(z - z0)n} is unbounded, while if p c R and we pick r satisfying p c r c R, then by definition of R we shall have that {Icnlr"}is a bounded sequence, say Ic,,Jr"I M < 00 for all n.

As is customary nowadays, the account to be presented here imperiously conscripts the best features from each of these. The strategy consists in exploiting first one (until it encounters a chasm it cannot bridge) and then another of these points of view. And, as with any anabasis, extensive logistical preparation is necessary. 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.

28 are examples; here is another: Theorem (POMPEIU[1905a], LOOMAN [1925], RIDDER[1930a]) r f f is continuous in U,&,,,,l(f(z + h) - f(z))/hl isfinite for all but countably many z in U and f’ exists almost everywhere in U,then f is holomorphic in U. For details of these results, which are heavily real-variable in nature, the reader may consult MENCHOFF [1936], pp. 195-201 of SAKS[1937] or the exhaustive (but terse) treatise of TROKHIMCHUK [1964]. For a somewhat more elementary account of somewhat weaker results in this direction, see MEIER[1951], [1960] and ARSOVE [1955].

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