# proving generalised euler's formula using elliptic functions

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?

• What exactly do you mean by "using the theory of elliptic functions"? Without defining what that means, it is impossible to give a good answer to your question. Please include this definition or a link to where this is defined in your question itself. Mar 7 at 0:20

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.
• Jacobi invented his theory of elliptic functions and that was his original definition. That is, sine amplitude $\sin\textrm{am}.(u|k) = \sin(am(u|k))$. I can look up the original equations in his "Fundamenta Nova Theoriae Functionum Ellipticarum" for you if you wish. Or just look in Jacobi Elliptic Function in Wikipedia. Aug 15, 2017 at 3:00
• @Nicco Then I suggest being explicit about that in your question, and being explicit what you consider the definition of $sn()$ and $cn()$ because there are multiple ways to define them. Aug 15, 2017 at 3:42