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Concerning transcendentally transcendental functions

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Literatur

  • Mr. Weber (in his memoir:Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie; Mathematische Annalen, vol. 43, pp. 521–549, November, 1893 —and in his book:Lehrbuch der Algebra, vol. 1, pp. 441–460, 1895) denotes by the termKörper any closed system whatever of say (Grössen, Elemente or, as I prefer)marks capable of combination without ambiguity by addition, subtraction, multiplication and division (division by 0 alone excluded) in accordance with the ordinaryoperational identities of algebra. I propose the termfield as the English equivalent toKörper. [Indeed thefield of my paper:A doubly infinite system of simple groups: presented August 25, 1893 to the Chicago Congress on Mathematics was exactly Mr. Weber'sendlicher Körper].

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  • Kronecker's term isRationalitätsbereich (realm of rationality). Mr. Weber (Algebra, vol. I, p. 452) proposes to use this term instead of Körper (field) only when in a particular investigation the marks of the field are to be considered as known or rational. This is surely a desirable usage, and I follow it in using in these papers:Concerning transcendentally transcendental functions: the term realm of rationality.

  • For the fundamental theorems concerning forms one may consult § 3 of Mr. Weber's memoir cited above.

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  • See, for instance, for ϕ(x), Stäckel: Zur Theorie der eindeutigen analytischen Functionen; Journal für die reine und angewandte Mathematik, vol. 112, pp. 262–264, 1893.

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  • Jacobi: Ueber die Differentialgleichung, welcher die Reihen\(1 \pm 2q + 2q^4 \pm 2q^9 + etc.,2\sqrt[4]{q} + 2\sqrt[4]{{q^9 }} + 2\sqrt[4]{{q^{25} }} + etc.\) Genüge leisten; Journal für die reine und angewandte Mathematik, vol. 36, pp. 97–112, 1834, or Gesammelte Werke, vol. II, pp. 173–190.

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  • Communicated to Mr. Schwarz by Kronecker in 1863. Schwarz: Ueber diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt. Journal für die reine und angewandte Mathematik, vol. 75, pp. 292–335, 1873, or Gesammelte Mathematische Abhandlungen, vol. II, p. 211–259. (See page 241.)

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  • Méray: Sur l'impossibilité de franchir par la formule de Taylor les cercles de convergence de certaines séries entières: Bulletin des Sciences mathématiques, series 2, vol. 12, pp. 248–252, 1888.

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    Eliakim Hastings Moore

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Moore, E.H. Concerning transcendentally transcendental functions. Math. Ann. 48, 49–74 (1896). https://doi.org/10.1007/BF01446334

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