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,

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

See also Weierstrass elliptic functions; Elliptic function.

References

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

References

[a1]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) pp. Chapt. 13