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On a Certain Series of Abel

Published online by Cambridge University Press:  20 January 2009

J. Clunie
J. Clunie
Affiliation:
University College of North Staffordshire, Keele.
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If we express ezλ, qua function of λ, in terms of λeλ by Langrange's theorem we find, for |λ| sufficiently small,

If we now regard λ as a parameter and z as the independent variable then (1.1) can be written

Supposing, therefore, that a function is of the form then

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 9 , Issue 3 , July 1956 , pp. 132 - 144
Copyright
Copyright © Edinburgh Mathematical Society 1956

References

REFERENCES

1.Abel, N. H., Oeuvres d'Abel (1881), t. 2, 67–81.Google Scholar
2.Halphen, G. H., “Sur une serie d'Abel”, Bull, de la Soc. Math. (1881–1882), 67.Google Scholar
3.Hayes, N. D., “Roots of the transcendental equation associated with a certain difference-differential equation”, Journal London Math. Soc., 25 (1950), 226–232.CrossRefGoogle Scholar
4.Macintyre, A. J. and Macintyre, S. S., “Theorems on the convergence and asymptotic validity of Abel's series”, Proc. Roy. Soc. Edinburgh, A, 63 (1952), 222–231.Google Scholar