Lemniscate functions

A special case of elliptic functions (cf. Elliptic function). They arise in the inversion of the elliptic integral of special form

$$ z = \int\limits _ { 0 } ^ { u } ( 1 - t ^ {4} ) ^ {- 1/2} dt . $$

These integrals first appeared in the calculation of the arc length of the Bernoulli lemniscate in the work of G. Fagnano (1715). Lemniscate functions themselves were introduced by C.F. Gauss (1797).

There are two lemniscate functions:

$$ u = \cos \mathop{\rm lemn} z = \mathop{\rm cl} z $$

and

$$ \sin \mathop{\rm lemn} z = \mathop{\rm sl} z = \cos \mathop{\rm lemn} \left ( \frac \omega {2} - z \right ) , $$

where

$$ \frac \omega {2} = \int\limits _ { 0 } ^ { 1 } ( 1 - t ^ {4} ) ^ {- 1/2} dt = \frac{ \sqrt 2 }{8 \sqrt \pi } \left [ \Gamma \left ( \frac{1}{4} \right ) \right ] ^ {2} . $$

The lemniscate functions can be expressed in terms of the Jacobi elliptic functions with modulus $ k = \sqrt 2 / 2 $:

$$ \mathop{\rm sl} z = \frac{\sqrt 2 }{2} \frac{ \mathop{\rm sn} ( z \sqrt 2 ) }{ \mathop{\rm dn} ( z \sqrt 2 ) } ,\ \mathop{\rm cl} z = \mathop{\rm cn} ( z \sqrt 2 ) . $$

In the theory of Weierstrass elliptic functions the lemniscate functions occur in the so-called harmonic case, when the invariants $ g _ {2} = 4 $, $ g _ {3} = 0 $.

References

[1]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2