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:
“The real weakness of England lies, not in incomplete armaments or unfortified coasts, not in the poverty that creeps through sunless lanes, or the drunkenness that brawls in loathsome courts, but simply in the fact that her ideals are emotional and not intellectual.”
—Oscar Wilde (1854–1900)
“Self-centeredness is a natural outgrowth of one of the toddler’s major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, what’s his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of him—an integral piece—is being torn from him.”
—Lawrence Balter (20th century)
“It was easy to recognize in him the anti-social animus of a born evangelist, but there was also something else—a kind of voluptuous delight in the shabby and preposterous, a perverted aestheticism like that of a latter-day movie or radio fan, a wild will to roll in and snuffle balderdash as a cat rolls in and snuffles catnip.”
—H.L. (Henry Lewis)