Given the elliptic modulus $k$,such that the complementary modulus is defined by $k'\equiv \sqrt{1-k^2}$,where $\phi\equiv am(u|k)$ is the jacobi amplitude and $K(k)$ is the complete elliptic integral of the first kind.
The elliptic generalisation of euler's formula in complex analysis(which expresses jacobi's elliptic functions in terms of the exponential function) is
$e^{i \phi}=\text{cn(u|k)}+i\text{sn(u|k)}$
The formula is valid multiples of $4K(k)$ and also multivalued.
How would we prove the formula using the theory of elliptic functions instead of elliptic functions defined in terms of trigonometric functions?
By Jacboi's original definition, $\text{cn(u|k)}:=\cos\phi$ and $\text{sn(u|k)}:=\sin\phi$. But $e^{i\phi}=\cos\phi+i\sin\phi$ by Euler and the result follows.