Jacobi Elliptic Functions - Jacobi Elliptic Functions As Solutions of Nonlinear Ordinary Differential Equations

Jacobi Elliptic Functions As Solutions of Nonlinear Ordinary Differential Equations

The derivatives of the three basic Jacobi elliptic functions are:

\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{sn}\,(z) = \mathrm{cn}\,(z)\, \mathrm{dn}\,(z),



\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{dn}\,(z) = - k^2 \mathrm{sn}\,(z)\, \mathrm{cn}\,(z).

With the addition theorems above and for a given k with 0 < k < 1 they therefore are solutions to the following nonlinear ordinary differential equations:

  • solves the differential equations
and
  • solves the differential equations
and
  • solves the differential equations
and

Read more about this topic:  Jacobi Elliptic Functions

Famous quotes containing the words jacobi, functions, solutions, ordinary and/or differential:

    “... [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)

    “When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
    —Edward T. Hall (b. 1914)

    “Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
    —Rebecca West (1892–1983)

    “I reject all evidence that my fabulous beloved is an ordinary person who worries, watches TV, and has bouts of indigestion.”
    —Mason Cooley (b. 1927)

    “But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
    —Antonin Artaud (1896–1948)