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The Weierstrass half‐periods  and the Weierstrass invariants  , the Weierstrass  function values at half-periods  , and the Weierstrass zeta function values at half-periods  are defined by the following formulas:
where  is the Klein invariant modular function,  is Weierstrass elliptic  function, and  is theWeierstrass zeta function.
The previous four vector functions are sometimes called the Weierstrass utilities because they are the basic elements of the Weierstrass theory of elliptic functions.
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