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:

    “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)

    “The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
    —Susan Sontag (b. 1933)

    “... [the] special relation of women to children, in which the heart of the world has always felt there was something sacred, serves to impress upon women certain tendencies, to endow them with certain virtues ... which will render them of special value in public affairs.”
    —Mary Putnam Jacobi (1842–1906)

    “Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.”
    —Terri Apter (20th century)