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Composition Theorems on Dirichlet Series

  • M. D. Penry (a1)

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When two uniform functions , are given, each with a finite radius of absolute convergence R1 R2 respectively, and {λn}, {μν} are real positive increasing sequences tending to infinity, a theorem due to Eggleston [1], which is a generalisation of Hurwitz1 s composition theorem, gives information about the position of the singularities of a composition function h(z), which is assumed to be uniform, in terms of the position of the singularities of f(z) and g(z). This result can be extended to Dirichlet series with real exponents by use of the transformation z = es.

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References

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1. Eggleston, H. G., A generalization of the Hurwitz composition theorem to irregular power series, Proc. Cambridge Philos. Soc. 47 (1951), pp. 477-482.
2. Hille, E., Note on Dirichlet series with complex exponents, Ann. of Math. (2), 25, (1924) pp. 261-78.
3. Schwengeler, E., Geometrisches űber die Verteilung der Nullstellen spezieller ganzer Funktionen, Dissertation, Zurich (1925).
4. Whittaker, E. T. and Watson, G.N., A course of modern analysis, Cambridge (1902).
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Composition Theorems on Dirichlet Series

  • M. D. Penry (a1)

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