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

sn

{\displaystyle \operatorname {sn} }

for

sin

{\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, but his work was published much later.

Overview

There are twelve Jacobi elliptic functions denoted by

pq

⁡

(

u

,

m

)

{\displaystyle \operatorname {pq} (u,m)}

, where

p

{\displaystyle \mathrm {p} }

and

q

{\displaystyle \mathrm {q} }

are any of the letters

c

{\displaystyle \mathrm {c} }

,

s

{\displaystyle \mathrm {s} }

,

n

{\displaystyle \mathrm {n} }

, and

d

{\displaystyle \mathrm {d} }

. (Functions of the form

pp

⁡

(

u

,

m

)

{\displaystyle \operatorname {pp} (u,m)}

are trivially set to unity for notational completeness.)

u

{\displaystyle u}

is the argument, and

m

{\displaystyle m}

is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both

u

{\displaystyle u}

and

m

{\displaystyle m}

.