- #1
bamajon1974
- 21
- 5
I need help in understanding how Jacobian Elliptic Functions are interpreted as inverses of Elliptic Functions.
Please reference the wiki page on Jacobian Elliptic functions:
https://en.wikipedia.org/wiki/Jacobi_elliptic_functions
For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) = \int_0^φ \frac{1}{ \sqrt{1-m sin^2(θ)}} \, dθ$$
where ##φ## is the amplitude $$φ=am(u)$$
Then $$y'=sin(φ)$$ is defined to be the Jacobian function $$sn(u,m)$$ or $$sin(φ)=sn(u,m)$$
I understand that the Jacobian functions are defined as inverses to the elliptic integrals. Bug where does the ##sin(φ)## come from? I think I am getting getting lost with all of the different variables. Can someone explain in as clearly as possible? Any insight would be appreciated. Thanks!
Please reference the wiki page on Jacobian Elliptic functions:
https://en.wikipedia.org/wiki/Jacobi_elliptic_functions
For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) = \int_0^φ \frac{1}{ \sqrt{1-m sin^2(θ)}} \, dθ$$
where ##φ## is the amplitude $$φ=am(u)$$
Then $$y'=sin(φ)$$ is defined to be the Jacobian function $$sn(u,m)$$ or $$sin(φ)=sn(u,m)$$
I understand that the Jacobian functions are defined as inverses to the elliptic integrals. Bug where does the ##sin(φ)## come from? I think I am getting getting lost with all of the different variables. Can someone explain in as clearly as possible? Any insight would be appreciated. Thanks!