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:

    “... spinsterhood [is considered to be] an abnormality of small proportions and small consequence, something like an extra finger or two on the body, presumably of temporary duration, and never of any social significance.”
    —Mary Putnam Jacobi (1842–1906)

    “In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
    —Urie Bronfenbrenner (b. 1917)

    “Science fiction writers foresee the inevitable, and although problems and catastrophes may be inevitable, solutions are not.”
    —Isaac Asimov (1920–1992)

    “How far men go for the material of their houses! The inhabitants of the most civilized cities, in all ages, send into far, primitive forests, beyond the bounds of their civilization, where the moose and bear and savage dwell, for their pine boards for ordinary use. And, on the other hand, the savage soon receives from cities iron arrow-points, hatchets, and guns, to point his savageness with.”
    —Henry David Thoreau (1817–1862)

    “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)