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:
“Each of us is incomplete compared to someone else, an animal’s incomplete compared to a person ... and a person compared to God, who is complete only to be imaginary.”
—Georges Bataille (1897–1962)
“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)
“Thoughts tending to content flatter themselves
That they are not the first of fortune’s slaves,
Nor shall not be the last, like silly beggars
Who, sitting in the stocks, refuge their shame
That many have and others must sit there,
And in this thought they find a kind of ease.”
—William Shakespeare (1564–1616)