Jacobi elliptic functions

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, 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 $\operatorname {sn}$ 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). 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.

^ Armitage, J. V.; Eberlein, W. F. (2006). Elliptic Functions (First ed.). Cambridge University Press. ISBN 978-0-521-78078-0. p. 48

[1] Armitage, J. V.; Eberlein, W. F. (2006). Elliptic Functions (First ed.). Cambridge University Press. ISBN 978-0-521-78078-0. p. 48

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