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On the Definitions of Elliptic Functions

Published online by Cambridge University Press:  03 November 2016

C. A. B. Smith
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Extract

In practice it is generally as inverse functions of elliptic integrals that elliptic functions are needed; and so a simple demonstration of this important property of the functions is to be desired. The important step in such a demonstration is a proof that the inverse of an elliptic integral is a one-valued function, which for every value of the variable is regular or has at worst a pole—that is, is meromorphic over the complete plane (save possibly at ∞).

Type
Research Article
Information
The Mathematical Gazette , Volume 28 , Issue 279 , May 1944 , pp. 41 - 45
Copyright
Copyright © Mathematical Association 1944

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References

Page 41 of note * Goursat, Cours d’analyse mathématique, vol. ii. 3rd ed. (1918), p. 536 (chap. xxi); E. H. Neville, Jacobian elliptic functions, pp. 137–139.

Page 44 of note * We do not go into this in detail here, as in a letter Prof. Neville has said that there is a full discussion in his new book, Jacobian Elliptic Functions. See in addition M. M. U. Wilkinson, “ Elliptic and Allied Functions ”, Proc. 5th Inter. Cong. Math. (1912), vol. i, p. 407

Page 44 of note † For detailed proofs see E. H. Neville, Jacobian Elliptic Functions.