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Evaluating values of the Weierstrass -function

asked 11 years ago

Blackadder gravatar image

I would like to know how can we evaluate the Weierstrass -functions. That is, I would like to find (θ,ω,iω) for some θ,ωR.

I'm only able to find a function which outputs the Laurent series of the Weierstrass -function when an elliptic curve has been entered. Should I evaluate that laurent series?

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answered 11 years ago

Luca gravatar image

The function weierstrass_p() of EllipticCurve returns the Laurent expansion of at the origin, hence evaluating it only gives a reasonable approximation near it.

Nothing comes to my mind to do this kind of numerical evaluation straightforwardly in Sage. If you have access to Maple, you could give a shot at the NumGFun package, part of the AlgoLib library http://algo.inria.fr/libraries/. It has support for the numerical evaluation of functions satisfying linear differential equations with polynomial coefficients.

The author of NumGFun gave a talk on it at Sage Days 49 http://www.marc.mezzarobba.net/#expose-sd49. As you can read in the slides, nothing of it is already in Sage, but there are plans for the close future. See, for example, http://trac.sagemath.org/ticket/14996, which will add functionality similar to what you need.

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The Jacobi elliptic functions in #14996 are already in Sage, the patch just improves them.

Eviatar Bach gravatar imageEviatar Bach ( 11 years ago )

Nice, I didn't know that. Then you may try using the formulas given here: <http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Relation_to_Jacobi_elliptic_functions></p<>>

Luca gravatar imageLuca ( 11 years ago )

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Asked: 11 years ago

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Last updated: Aug 07 '13