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{{Short description|Mathematical function}}
In [[mathematics]], the '''Jacobi elliptic functions''' are a set of basic [[elliptic function]]s. They are found in the description of the [[pendulum (mechanics)|motion of a pendulum]], as well as in the design of electronic [[elliptic filter]]s. While [[trigonometry|trigonometric functions]] are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other [[conic section]]s, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation <math>\operatorname{sn}</math> for <math>\sin</math>. 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 {{harvs|txt|first=Carl Gustav Jakob |last=Jacobi|authorlink=Carl Gustav Jakob Jacobi|year=1829}}. [[Carl Friedrich Gauss]] had already studied special Jacobi elliptic functions in 1797, the [[lemniscate elliptic functions]] in particular,<ref>{{Cite book |last1=Armitage |first1=J. V. |last2=Eberlein| first2=W. F. |title=Elliptic Functions |publisher=Cambridge University Press |year=2006 |edition=First |isbn=978-0-521-78078-0}} p. 48</ref> but his work was published much later.
==Overview==
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There are twelve Jacobi elliptic functions denoted by <math>\operatorname{pq}(u, m)</math>, where <math>\mathrm p</math> and <math>\mathrm q</math> are any of the letters <math>\mathrm c</math>, <math>\mathrm s</math>, <math>\mathrm n</math>, and <math>\mathrm d</math>. (Functions of the form <math>\operatorname{pp}(u,m)</math> are trivially set to unity for notational completeness.) <math>u</math> is the argument, and <math>m</math> is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are [[meromorphic function|meromorphic]] in both <math>u</math> and <math>m</math>.<ref name="Walker">{{cite journal |last1=Walker |first1=Peter |date=2003 |title=The Analyticity of Jacobian Functions with Respect to the Parameter k |url=https://www.jstor.org/stable/3560143 |bibcode=<!-- useless bibcode 2003RSPSA.459.2569W --> |journal=Proceedings of the Royal Society |volume=459 |issue=2038 |pages=2569–2574|doi=10.1098/rspa.2003.1157 |jstor=3560143 }}</ref> The distribution of the zeros and poles in the <math>u</math>-plane is well-known. However, questions of the distribution of the zeros and poles in the <math>m</math>-plane remain to be investigated.<ref name="Walker"/>
In the complex plane of the argument <math>u</math>, the twelve functions form a repeating lattice of simple [[Zeros and poles|poles and zeroes]].<ref name="DLMF22">{{cite web|url=http://dlmf.nist.gov/22|title=NIST Digital Library of Mathematical Functions (Release 1.0.17)|editor-last=Olver|editor-first=F. W. J.|display-editors=et al |date=2017-12-22|publisher=National Institute of Standards and Technology|access-date=2018-02-26 }}</ref> Depending on the function, one repeating [[parallelogram]], or unit cell, will have sides of length <math>2K</math> or <math>4K</math> on the real axis, and <math>2K'</math> or <math>4K'</math> on the imaginary axis, where <math>K=K(m)</math> and <math>K'=K(1-m)</math> are known as the [[quarter period]]s with <math>K(\cdot)</math> 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 <math>(0,0)</math> at one corner, and <math>(K,K')</math> as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named <math>\mathrm s</math>, <math>\mathrm c</math>, <math>\mathrm d</math>, and <math>\mathrm n</math>, going counter-clockwise from the origin. The function <math>\operatorname{pq}(u,m)</math> will have a zero at the <math>\mathrm p</math> corner and a pole at the <math>\mathrm q</math> corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.
When the argument <math>u</math> and parameter <math>m</math> are real, with <math>0 < m < 1</math>, <math>K</math> and <math>K'</math> will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.
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&=\frac{\pi}{2K}\frac{\theta_{2}'(\zeta|\tau)}{\theta_{2}(\zeta|\tau)}+\frac{\operatorname{dn}(u,m)\operatorname{sn}(u,m)}{\operatorname{cn}(u,m)}\\
&=\frac{\pi}{2K}\frac{\theta_{1}'(\zeta|\tau)}{\theta_{1}(\zeta|\tau)}-\frac{\operatorname{cn}(u,m)\operatorname{dn}(u,m)}{\operatorname{sn}(u,m)}\end{align}</math>
where <math>'</math> denotes the [[partial derivative]] with respect to the first variable.
===Using modular inversion===
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<math display="block">\operatorname{sn}\left(\frac{u}{2},m\right)=\pm\sqrt{\frac{1-\operatorname{cn}(u,m)}{1+\operatorname{dn}(u,m)}}</math>
<math display="block">\operatorname{cn}\left(\frac{u}{2},m\right)=\pm\sqrt{\frac{\operatorname{cn}(u,m)+\operatorname{dn}(u,m)}{1+\operatorname{dn}(u,m)}}</math>
<math display="block">\operatorname{
===K formulas===
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