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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. Many of their features show up in important structures and have direct relevance to some applications (e.g. the equation of a pendulum—also see pendulum (mathematics)). They also have useful analogies to the functions of trigonometry, as indicated by the matching notation ''sn'' for ''sin''. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . ==Introduction== 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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Jacobi elliptic functions」の詳細全文を読む スポンサード リンク
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