Jacobian function

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of the electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, 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 Carl Gustav Jakob Jacobi (1829).

There are twelve Jacobi elliptic functions denoted by pq(u,m), where p and q are any of the letters c, s, n, and d. (Functions of the form pp(u,m) are trivially set to unity for notational completeness). u is the argument, and m is the parameter, both of which may be complex.

In the complex plane of the argument u, the twelve functions form a repeating rectangular lattice of simple poles and zeroes. Depending on the function, one repeating rectangle, or unit cell, will have sides of length 2K or 4K on the real axis, and 2K' or 4K' on the imaginary axis, where K=K(m) and K'=K(1-m) are known as the quarter periods with K(.) being the elliptic integral of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle", which is a rectangle formed by the origin (0,0) at one corner, and (K,K') as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named s, c, d, and n, going counter-clockwise from the origin. The function pq(u,m) will have a zero at the "p" corner and a pole at the "q" corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

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