# Delta amplitude

One of the three fundamental Jacobi elliptic functions. It is denoted by

$${ \mathop{\rm dn} } u = { \mathop{\rm dn} } ( u , k ) = \Delta { \mathop{\rm am} } u .$$

The delta amplitude is expressed as follows in terms of the Weierstrass sigma-function, the Jacobi theta-functions or a series:

$${ \mathop{\rm dn} } u = { \mathop{\rm dn} } ( u , k ) = \ \frac{\sigma _ {2} ( u) }{\sigma _ {3} ( u) } = \ \frac{\theta _ {0} ( 0) \theta _ {3} ( v) }{\theta _ {3} ( 0) \theta _ {0} ( v) } =$$

$$= \ 1 - k ^ {2} \frac{u ^ {2} }{2!} + k ^ {2} ( 4 + k ^ {2} ) \frac{u ^ {4} }{4!} - k ^ {2} ( 16 + 44k ^ {2} + k ^ {4} ) \frac{u ^ {6} }{6!} - \dots ,$$

where $k$ is the modulus of the delta amplitude, $0 \leq k \leq 1$, and $v = u /2 \omega$, $2 \omega = \pi ( \theta _ {3} ( 0)) ^ {2}$. If $k= 0, 1$ one has, respectively,

$${ \mathop{\rm dn} } u = 1 ,\ \ { \mathop{\rm dn} } u = \frac{1}{\cosh u } .$$

#### References

 [1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) pp. Chapt. 3, Abschnitt 2