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)
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (1533–1592)
“So long as the law considers all these human beings, with beating hearts and living affections, only as so many things belonging to the master—so long as the failure, or misfortune, or imprudence, or death of the kindest owner, may cause them any day to exchange a life of kind protection and indulgence for one of hopeless misery and toil—so long it is impossible to make anything beautiful or desirable in the best-regulated administration of slavery.”
—Harriet Beecher Stowe (1811–1896)