I found this identity in the wiki(adsbygoogle = window.adsbygoogle || []).push({});

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)}##

?

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# Complementary jacobi amplitude

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