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:
“A man in love is incomplete until he has married—then he’s finished.”
—Zsa Zsa Gabor (b. 1919)
“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)
“... wariness about change is a kind of prairie wisdom.”
—Kathleen Norris (b. 1947)