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 e1-3 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}}.

Read more about this topic:  Weierstrass's Elliptic Functions

Famous quotes containing the words relation to, relation, jacobi and/or functions:

    “Among the most valuable but least appreciated experiences parenthood can provide are the opportunities it offers for exploring, reliving, and resolving one’s own childhood problems in the context of one’s relation to one’s child.”
    —Bruno Bettelheim (20th century)

    “There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
    —Adrienne Rich (b. 1929)

    “During the long ages of class rule, which are just beginning to cease, only one form of sovereignty has been assigned to all men—that, namely, over all women. Upon these feeble and inferior companions all men were permitted to avenge the indignities they suffered from so many men to whom they were forced to submit.”
    —Mary Putnam Jacobi (1842–1906)

    “Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
    —Kate Millett (b. 1934)