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On the periods of the exponential and elliptic functions

Published online by Cambridge University Press:  24 October 2008

D. W. Masser
Affiliation:
Trinity College, Cambridge

Extract

Let ℘ be a Weierstrass elliptic function satisfying the differential equation

and let ζ(z) be the associated Weierstrass ζ-function satisfying (z) = −℘(z). Corresponding to a pair of fundamental periods ω1, ω2 of ℘(z), there is a pair of quasi-periods η1, η2 of ζ(z) defined by

and we have ηi = 2ζ(ωi/2) for i = 1, 2.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

(1)Baker, A.An estimate for the ℘-function at an algebraic point, Amer. J. Math. 92 (1970), 619622.CrossRefGoogle Scholar
(2)Baker, A.On the quasi-periods of the Weierstrass ζ-function, Nachr. Akad. Wiss. Göttingen Math-Phys. Kl. II (1969), 145157.Google Scholar
(3)Coates, J.An application of the division theory of elliptic functions to diophantine approximation, Invent. Math. 11 (1970), 167182.CrossRefGoogle Scholar
(4)Coates, J.Construction of rational functions on a curve. Proc. Cambridge Philos. Soc. 68 (1970), 105123.CrossRefGoogle Scholar
(5)Coates, J.The transcendence of linear forms in ω1, ω2, ηl, η2, 2πi. Amer. J. Math. 93 (1971), 385397.CrossRefGoogle Scholar
(6)Fel'dman, N. I.An elliptic analogue of an inequality of A. O. Gel'fond. Trans. Moscow Math. Soc. 18 (1968), 7184.Google Scholar
(7)Weyl, H.Algebraic theory of numbers. Ann. of Math. Studies 1 (Princeton, 1940).Google Scholar
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On the periods of the exponential and elliptic functions
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