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Elliptic Functions by W. F. Eberlein, J. V. Armitage

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7 The Weierstrass elliptic functions

The treatment offered here owes much to lectures by A. M. Macbeath.

7.1 Construction of the Weierstrass functions; the Weierstrass sigma and zeta functions; the Weierstrass U+2118-function

At the end of Chapter 3 we made brief reference to the possibility of constructing an elliptic function having periods (ω1, ω2) and a double pole at the origin, with residue 0. We assumed its existence and anticipated its development,

℘⁡(z)=z−2+∑ω≠0{(z−ω)−2−ω−2},
(7.1)

where ω runs through the points ω = n1ω1 + n2ω2 of the period lattice, Λ, other than the point ...

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