Elliptic Integral - Incomplete Elliptic Integral of The First Kind

The incomplete elliptic integral of the first kind F is defined as

This is the trigonometric form of the integral; substituting, one obtains Jacobi's form:

Equivalently, in terms of the amplitude and modular angle one has:

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude:

This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

With one has:

thus, the Jacobian elliptic functions are inverses to the elliptic integrals.

Read more about this topic:  Elliptic Integral

Famous quotes containing the words incomplete, integral and/or kind:

    Someone once asked me why women don’t gamble as much as men do, and I gave the common-sensical reply that we don’t have as much money. That was a true but incomplete answer. In fact, women’s total instinct for gambling is satisfied by marriage.
    Gloria Steinem (b. 1934)

    An island always pleases my imagination, even the smallest, as a small continent and integral portion of the globe. I have a fancy for building my hut on one. Even a bare, grassy isle, which I can see entirely over at a glance, has some undefined and mysterious charm for me.
    Henry David Thoreau (1817–1862)

    There seems to be a kind of order in the universe, in the movement of the stars and the turning of the earth and the changing of the seasons, and even in the cycle of human life. But human life itself is almost pure chaos. Everyone takes his stance, asserts his own rights and feelings, mistaking the motives of others, and his own.
    Katherine Anne Porter (1890–1980)