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I have just learned the definition of the first Jacobian elliptic function $\text{sn}(u)=\text{sn}(u,k)$, defined as the sine of the inverse function of $$F(\phi,k):=\int_0^\phi \frac{dx}{\sqrt{1-k^2\sin^2(x)}}$$ with respect to $\phi$; that is, if $u=F(\phi,k)$, then $\text{sn}(u,k)=\sin\phi$. I also know the analogous definitions of $\text{cn}(u,k)$ and $\text{dn}(u,k)$.

QUESTION. I have seen the following addition formula and would like to know how to prove it: $$\text{sn}(u+v)=\frac{\text{sn}(u)\text{cn}(v)\text{dn}(v)+\text{sn}(v)\text{cn}(u)\text{dn}(u)}{1-k^2\text{sn}^2(u)\text{sn}^2(v)}$$ How is this formula proven? Does it require any "advanced machinery" that a beginner would not be familiar with, or is it a simple proof?

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  • $\begingroup$ Maybe not with all the details you are asking for but pages 4,5,6,7,8 of this seem to be helpful. It looks like there is a result which is true of a broader class of functions and this is just an application of this. $\endgroup$ – Mason Nov 15 '18 at 22:05
  • $\begingroup$ @Mason Wow, that's great! That is exactly what I was looking for. Thank you so much! $\endgroup$ – Frpzzd Nov 15 '18 at 22:09
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    $\begingroup$ @Mason Well, if I can work out the details then I will post it as an answer to my own question. $\endgroup$ – Frpzzd Nov 15 '18 at 22:27
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    $\begingroup$ This is a very standard result and is presented in most textbooks which cover elliptic functions. You may also have a look at my post based on Cayley's book. $\endgroup$ – Paramanand Singh Nov 21 '18 at 15:14
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    $\begingroup$ In general no advanced machinery is needed to understand elliptic functions unless you want to study its link with algebraic number theory. For me this link has been so notoriously difficult. $\endgroup$ – Paramanand Singh Nov 21 '18 at 15:17

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