In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of 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 ${\displaystyle \operatorname {sn} }$ for ${\displaystyle \sin }$. 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). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular,^[1] but his work was published much later.

Overview

There are twelve Jacobi elliptic functions denoted by ${\displaystyle \operatorname {pq} (u,m)}$, where ${\displaystyle \mathrm {p} }$ and ${\displaystyle \mathrm {q} }$ are any of the letters ${\displaystyle \mathrm {c} }$, ${\displaystyle \mathrm {s} }$, ${\displaystyle \mathrm {n} }$, and ${\displaystyle \mathrm {d} }$. (Functions of the form ${\displaystyle \operatorname {pp} (u,m)}$ are trivially set to unity for notational completeness.) ${\displaystyle u}$ is the argument, and ${\displaystyle m}$ is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both ${\displaystyle u}$ and ${\displaystyle m}$.^[2] The distribution of the zeros and poles in the ${\displaystyle u}$-plane is well-known. However, questions of the distribution of the zeros and poles in the ${\displaystyle m}$-plane remain to be investigated.^[2]

In the complex plane of the argument ${\displaystyle u}$, the twelve functions form a repeating lattice of simple poles and zeroes.^[3] Depending on the function, one repeating parallelogram, or unit cell, will have sides of length ${\displaystyle 2K}$ or ${\displaystyle 4K}$ on the real axis, and ${\displaystyle 2K'}$ or ${\displaystyle 4K'}$ on the imaginary axis, where ${\displaystyle K=K(m)}$ and ${\displaystyle K'=K(1-m)}$ are known as the quarter periods with ${\displaystyle K(\cdot )}$ being the elliptic integral of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle" (generally a parallelogram), which is a rectangle formed by the origin ${\displaystyle (0,0)}$ at one corner, and ${\displaystyle (K,K')}$ as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named ${\displaystyle \mathrm {s} }$, ${\displaystyle \mathrm {c} }$, ${\displaystyle \mathrm {d} }$, and ${\displaystyle \mathrm {n} }$, going counter-clockwise from the origin. The function ${\displaystyle \operatorname {pq} (u,m)}$ will have a zero at the ${\displaystyle \mathrm {p} }$ corner and a pole at the ${\displaystyle \mathrm {q} }$ corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

When the argument ${\displaystyle u}$ and parameter ${\displaystyle m}$ are real, with ${\displaystyle 0<m<1}$, ${\displaystyle K}$ and ${\displaystyle K^{\prime }}$Zdroj:https://en.wikipedia.org?pojem=Jacobi_elliptic_functions
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