Jacobi Elliptic Functions - Addition Theorems

Addition Theorems

The functions satisfy the two algebraic relations

From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions

\begin{align} \operatorname{cn}(x+y) & = {\operatorname{cn}(x)\;\operatorname{cn}(y) - \operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{dn}(x)\;\operatorname{dn}(y) \over {1 - k^2 \;\operatorname{sn}^2 (x) \;\operatorname{sn}^2 (y)}}, \\ \operatorname{sn}(x+y) & = {\operatorname{sn}(x)\;\operatorname{cn}(y)\;\operatorname{dn}(y) + \operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{dn}(x) \over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}, \\ \operatorname{dn}(x+y) & = {\operatorname{dn}(x)\;\operatorname{dn}(y) - k^2 \;\operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{cn}(y) \over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}. \end{align}

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