The relation with the fundamental half-periods $\omega$ and $\omega^\prime$ from @abramowitz_handbook_1964 is $$\begin{aligned}
\omega_R & = \omega + \delta \omega^\prime,\\
\omega_C & = \omega^\prime,\end{aligned}$$ where $\delta = 0$ if $\Delta > 0$ and $\delta = 1$ if $\Delta < 0$. Since we are interested in the behaviour of $\wp$ on the real axis (as $\tau$ is a real-valued variable), we can then regard $\wp\left( \tau;g_2,g_3\right)$ as a singly-periodic real-valued function of period $2\omega_R$.

It follows then straightforwardly from eqs. – that $\xi\left( \tau \right)$ and $\eta\left( \tau \right)$ are both periodic in $\tau$ with periods that, in general, will be different. Conversely, from eq. , it follows immediately that $\phi\left( \tau \right)$ is not periodic. Indeed, $\phi\left( \tau \right)$ is a function of the form $$f\left(\tau\right)=A+B\tau+C_\xi\ln\frac{\sigma_\xi\left(\tau + a_\xi\right)}{\sigma_\xi\left(\tau + b_\xi\right)}
+C_\eta\ln\frac{\sigma_\eta\left(\tau + a_\eta\right)}{\sigma_\eta\left(\tau + b_\eta\right)},\label{eq:generic_f_sigma}$$ where $A$, $B$, $C$, $a$ and $b$ are constants. It is now interesting to note that, according to eqs. and , if $\xi$ and $\eta$ have real half-periods $\omega_{R,\xi}$ and $\omega_{R,\eta}$ such that $$\frac{\omega_{R,\xi}}{\omega_{R,\eta}}=\frac{n}{m},$$ with $n$ and $m$ coprime natural numbers (or, in other words, $\omega_{R,\xi}$ and $\omega_{R,\eta}$ are commensurable), then $d\phi/d\tau$ becomes a periodic function with period $T = 2m\omega_{R,\xi} = 2n\omega_{R,\eta}$. Recalling the 