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On the periods of the exponential and elliptic functions

  • D. W. Masser (a1)

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.

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References

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(1)Baker, A.An estimate for the ℘-function at an algebraic point, Amer. J. Math. 92 (1970), 619622.
(2)Baker, A.On the quasi-periods of the Weierstrass ζ-function, Nachr. Akad. Wiss. Göttingen Math-Phys. Kl. II (1969), 145157.
(3)Coates, J.An application of the division theory of elliptic functions to diophantine approximation, Invent. Math. 11 (1970), 167182.
(4)Coates, J.Construction of rational functions on a curve. Proc. Cambridge Philos. Soc. 68 (1970), 105123.
(5)Coates, J.The transcendence of linear forms in ω1, ω2, ηl, η2, 2πi. Amer. J. Math. 93 (1971), 385397.
(6)Fel'dman, N. I.An elliptic analogue of an inequality of A. O. Gel'fond. Trans. Moscow Math. Soc. 18 (1968), 7184.
(7)Weyl, H.Algebraic theory of numbers. Ann. of Math. Studies 1 (Princeton, 1940).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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