I am trying to understand the how Jacobian Elliptic functions are derived from the corresponding elliptic integrals.
From http://mathworld.wolfram.com/JacobiEllipticFunctions.html on Jacobi Elliptic Functions, an elliptic integral of the first kind is:
$$u = F(\phi,k) = \int_0^\phi \frac{d\theta} {\sqrt{(1-k^2\sin^2\theta)}}$$
where
$$ 0<k^2<1 $$ and $$ \phi = am(u,k) $$
k is the modulus and am is the amplitude.
I know that the Jacobian elliptic functions are found be inverting the elliptic integrals.
If $$ u=F(\phi,k) $$
then the inverse is
$$ F^{-1}(u,k) = F^{-1}[F(\phi,k)] = \phi $$
and
$$ F^{-1}(u,k) = \phi = am(u,k) $$
So $$ \sin(\phi) = \sin(am(u,k)) = sn(u,k) $$
and
$$ \cos(\phi) = \cos(am(u,k)) = cn(u,k) $$
and $$\sqrt{1-k^2\sin^2\phi} = \sqrt{1-k^2\sin^2(am(u,k))} = dn(u,k)$$
Am I correct in my reasoning for inverting the integral? It looks like the elliptic integrals are functions of two variables (k and $\phi$), so when you invert the integral, what happens to the k? If $$\phi=am(u,k)$$ why to you take the sine of $\phi$?
Much thanks for any insight!
$k$ is normally regarded as a parameter in the integral, at least initially, with $\phi$ being the actual variable. The notation is odd for historic reasons, but one could equally write $\operatorname{am}_k(u)$ and so on. Think of what these functions are used for: solutions to differential equations like $$ u'' = -k^2\sin{u} $$ (although this is solved by am itself, rather than its sine or cosine), for example.