Digital Library of Mathematical Functions
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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.7 Quarter Periods

23.7.1 (12ω1) =e1+(e1-e3)(e1-e2)
=e1+ω1-2(K(k))2k,
23.7.2 (12ω2) =e2-(e1-e2)(e2-e3)
=e2-ω1-2(K(k))2kk,
23.7.3 (12ω3) =e3-(e1-e3)(e2-e3)
=e3-ω1-2(K(k))2k,

where k,k and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.