Weierstrass's Elliptic Functions - Relation To Jacobi Elliptic Functions

Relation To Jacobi Elliptic Functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of the Jacobi's elliptic functions. The basic relations are

$\wp(z) = e_{3} + \frac{e_{1} - e_{3}}{\mathrm{sn}^{2}\,w} = e_{2} + \left( e_{1} - e_{3} \right) \frac{\mathrm{dn}^{2}\,w}{\mathrm{sn}^{2}\,w} = e_{1} + \left( e_{1} - e_{3} \right) \frac{\mathrm{cn}^{2}\,w}{\mathrm{sn}^{2}\,w}$

where e_1-3 are the three roots described above and where the modulus k of the Jacobi functions equals

$k \equiv \sqrt{\frac{e_{2} - e_{3}}{e_{1} - e_{3}}}$

and their argument w equals

$w \equiv z \sqrt{e_{1} - e_{3}}.$

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