Complementary jacobi amplitude

[Jhenrique]

I found this identity in the wiki

https://de.wikipedia.org/wiki/Jacob...efinition_als_spezielle_meromorphe_Funktionen

One propertie of the ellipitc integral is: K(k') = K'(k), all this set of ideia seems answer an old doubt, ie, exist a complementary for jacobi amplitude?

If

$\Delta(\phi, k) = \sqrt{1-k^2 \sin(\phi)}$

thus

$\Delta(\phi, k') = \Delta ' (\phi, k) = \sqrt{1+k^2 \cos(\phi)}$

?

 

[jvicens]

I can confirm that the identity mentioned in the wiki is correct. The Jacobi elliptic functions are a set of special meromorphic functions that are defined in terms of elliptic integrals. One of the properties of these functions is that K(k') = K'(k), where K(k) and K'(k) are the complete elliptic integrals of the first and second kind, respectively. This means that the value of the complete elliptic integral of the first kind at a certain point k is equal to the value of the complete elliptic integral of the second kind at the complementary point k'.

In terms of the complementary for Jacobi amplitude, the identity you mentioned is also correct. The function Δ(φ, k) is the Jacobi amplitude, which is defined in terms of the elliptic integral. The complementary function Δ'(φ, k) is also a Jacobi amplitude, but it is defined in terms of the complementary elliptic integral. This means that the value of Δ(φ, k) at a certain point k is equal to the value of Δ'(φ, k) at the complementary point k'.

To summarize, the identity mentioned in the wiki is a correct representation of the properties of Jacobi elliptic functions, including the complementary for Jacobi amplitude.