Jacobi elliptic functions with modulus on the unit circle

Question

I am gathering some available informations on Jacobi elliptic functions $sn(z,k)$, $cn(z,k)$, $dn(z,k)$ with $k\in\mathbb{C}$, $|k|=1$. I can not find much on them in standard references (Abramowitz&Stegun, DLMF, wiki,...). It seems that something interesting happens in the particular regime when $|k|=1$.

For instance, I am interested in questions like: Is the formula for the Fourier series [Abramowitz&Stegun, eq. 16.23.2] $$ cn(z,k)=\frac{2\pi}{k K(k^{2})}\sum_{n=0}^{\infty}\frac{q^{n+1/2}}{1+q^{2n+1}}\cos\frac{(2n+1)\pi z}{2K(k^{2})} $$ still valid if $|k|=1$ and $k\neq\pm1$? What happens with the nome $q$? Is, for example, true that $|q|<1$? Etc.

Do you know some source where these properties of Jacobi elliptic functions with $|k|=1$ have been studied? Thanks!

Answer 1

Actually the only relevant case is the equianharmonic case, that's when $k=e^{i\pi/3}$ and $$K'(e^{i\pi/3})=e^{i\pi/6}K(e^{i\pi/3})=\frac{\pi^{1/2}\Gamma(1/6)}{{2}{\cdot}{3^{3/4}}\Gamma(2/3)}{\cdot}e^{-i\pi/6}$$The others are degenerate. Consult [1] page 306 Section 13.8 and page 320 Section 13.8.

References:
[1] A. ERDELYI, Editor. W. MAGNUS, F. OBERHETTINGER, F. G. TRICOMI, Research Associates. HIGHER TRANSCENDENTAL FUNCTIONS, vol. II
apps.nrbook.com/bateman/Vol2.pdf