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:

    “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)

    “The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
    —Cyril Connolly (1903–1974)

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

    “Certainly, then, ordinary language is not the last word: in
    principle it can everywhere be supplemented and improved upon, and superseded. Only remember, it is the first word.”
    —John Austin (1911–1960)

    “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)