In [[mathematics]], the '''Jacobi elliptic functions''' are a set of basic [[elliptic function]]s. They are found in the description of the motion of a [[pendulum]] (seemechanics)|motion alsoof a [[pendulum (mathematics)]]), 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.
