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Composition Theorems on Dirichlet Series

Published online by Cambridge University Press:  20 November 2018

M. D. Penry
M. D. Penry*
Affiliation:
University College of Wales, Aberystwyth
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 3 , 01 September 1963 , pp. 397 - 403
Copyright
Copyright © Canadian Mathematical Society 1963

References

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