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Math Help - Jacobi elliptic functions : the addition theorem

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    MHF Contributor Bruno J.'s Avatar
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    Jacobi elliptic functions : the addition theorem

    Hello!
    Now I know this topic is not the most popular in this century, and most math students today don't ever really learn about it. However I'm hoping somebody is familiar enough with Jacobi's theory as to help me solve this problem :

    Prove the addition theorem

    \mbox{sn}(u+v) =  \frac{\mbox{sn}(u) \mbox{cn}(v) \mbox{dn}(v)+\mbox{sn}(v)  \mbox{cn}(u) \mbox{dn}(u)}{1-k^2\mbox{sn}(u)^2\mbox{sn}(v)^2}.

    The hint given is : If s_1 =\mbox{sn}(u), s_2 =\mbox{sn}(v) and u+v=C is constant, then

    \frac{d}{du}\mbox{Ln}(s_1's_2-s_2's_1)=\frac{d}{du}\mbox{Ln}(1-k^2s_1^2s_2^2),


    where the primes denotes differentiation with respect to u, and hence \frac{s_1's_2-s_2's_1}{1-k^2s_1^2s_2^2} = \mbox{ constant }. Then prove that this constant equals \mbox{sn}(u+v).


    I can see how the addition theorem follows from the hint, but I'm having problem showing the two logarithmic derivatives above are in fact equal; the calculations get very messy and I'm not sure which of the many identities I should be using to simplify them.

    I also thought of proving the group law by using \mbox{sn} to parametrize the elliptic curve w^2=(1-z^2)(1-k^2z^2) and then adding the points on the curve using the group law. This might prove to be complicated and I'd like to know how to use the hint instead.

    Thanks a million!
    Last edited by Bruno J.; April 13th 2010 at 12:14 PM.
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