Jacobi's Elliptic Functions
There are twelve Jacobian elliptic functions. Each of the twelve corresponds to an arrow drawn from one corner of a rectangle to another. The corners of the rectangle are labeled, by convention, s, c, d and n. The rectangle is understood to be lying on the complex plane, so that s is at the origin, c is at the point K on the real axis, d is at the point K + iK' and n is at point iK' on the imaginary axis. The numbers K and K' are called the quarter periods. The twelve Jacobian elliptic functions are then pq, where each of p and q is one of the letters s, c, d, n.
The Jacobian elliptic functions are then the unique doubly periodic, meromorphic functions satisfying the following three properties:
- There is a simple zero at the corner p, and a simple pole at the corner q.
- The step from p to q is equal to half the period of the function pq u; that is, the function pq u is periodic in the direction pq, with the period being twice the distance from p to q. The function pq u is also periodic in the other two directions, with a period such that the distance from p to one of the other corners is a quarter period.
- If the function pq u is expanded in terms of u at one of the corners, the leading term in the expansion has a coefficient of 1. In other words, the leading term of the expansion of pq u at the corner p is u; the leading term of the expansion at the corner q is 1/u, and the leading term of an expansion at the other two corners is 1.
More generally, there is no need to impose a rectangle; a parallelogram will do. However, if K and iK' are kept on the real and imaginary axis, respectively, then the Jacobi elliptic functions pq u will be real functions when u is real.
Read more about this topic: Elliptic Function
Famous quotes containing the words jacobi and/or functions:
“During the long ages of class rule, which are just beginning to cease, only one form of sovereignty has been assigned to all men—that, namely, over all women. Upon these feeble and inferior companions all men were permitted to avenge the indignities they suffered from so many men to whom they were forced to submit.”
—Mary Putnam Jacobi (1842–1906)
“Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.”
—Terri Apter (20th century)