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A note on a theorem of Szegö. (English) JFM 63.0253.01

Ausgehend von einem bekannten Satz von G. Szegö (S. B. Preuß. Akad. Wiss. Phys.-math. Kl. 1922, 88-91; F. d. M. 48, 330 (JFM 48.0330.*)) über Potenzreihen mit nur endlich vielen verschiedenen Koeffizienten gewinnt Verf. eine Aussage über Lage und Art der auf dem Konvergenzkreis eventuell vorhandenen “dominierenden” Singularitäten von algebraisch-logarithmischem Typus (vgl. die nachstehend besprochene Arbeit) bei Potenzreihen, deren Koeffizientenfolge nur endlich viele verschiedene Häufungspunkte besitzt.

Citations:

JFM 48.0330.*
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References:

[1] Sitzungsberichte der Preußischen Akademie der Wissenschaften (1922), S. 88–91.
[2] A function is said to have an algebraic-logarithmic singularity at the pointz=z’ if it can be represented in the neighbourhood of this point by the sum of a finite number of terms of the form 540-1, wheres is complex,k a non-negative integer and {\(\psi\)}(z) is regular and non-zero atz=z’. The expression given is said to be oftype (s, k). Ifs, , ,... theweight of the element is [{\(\sigma\)},k], whereR(s)={\(\sigma\)}; ifs, , ,... andk>0, the weight is [s, k], while the weight of a regular point is [- 0]. The weight [{\(\sigma\)},k] is said to be greater than the weight [540-2,k’] if either 540-3 or 540-4 andk>k’. The weight of a singularity is defined to be the greatest of the weights of the component elements. Vide R. Jungen, Commentarii Mathematici Helvetici3 (1931), p. 274. · Zbl 0003.11901 · doi:10.1007/BF01601817
[3] R. Wilson, Proceedings of the London Mathematical Society (2)42 (1936), p. 211.
[4] R. Wilson,loc. cit., p. 213.
[5] Ibid., pp. 211–212.
[6] R. Wilson,loc. cit. pp. 219–220.
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