Proof of $\left| \operatorname{cn}\left( x(1+ i) \mid m \right)\right|=1$ for $m=\frac{1}{2}$ and $x \in \mathbb{R}$ Let $\operatorname{cn}(u\mid m)$ be Gudermann's notation for the Jacobi elliptic function $\operatorname{cn}$. It is well known to be doubly periodic. For $0<m<1$ the two periods, $4 K(m)$ and $4 i K(1-m)$, are purely real and purely imaginary respectively. For any $n_1, n_2 \in \mathbb{Z}$:
$$
    \operatorname{cn}\left(u + 4 n_1 K(m) + 4 n_2 i K(1-m) \mid m \right) = \operatorname{cn}\left(u \mid m \right)
$$
and where $K(m)$ denotes the complete elliptic integral of the first kind (in that link $m=k^2$). 
For $m=\frac{1}{2}$, $f(x,y) = \operatorname{cn}(x+i y \mid m)$ is doubly periodic in a square with sidelength $4K\left(\frac{1}{2}\right) \approx 7.4163$.
By looking at Peirce quincuncial projection I noticed empirically that, for $x\in \mathbb{R}$, $f(x,x)$ is of unit magnitude:
$$
    \left| f(x,x) \right| = \left| \operatorname{cn}\left(x+i x \mid \frac{1}{2}\right) \right| = 1
$$

I am looking for either a proof, or a reference to one. 
Thank you.

Jacobi's imaginary transformation
$$ \mathrm{cn}(iu|m)=\frac{1}{\mathrm{cn}(u|m')},$$
and parity of $\mathrm{cn}(u)$ will do the job. $\blacksquare$
