fundamental notions[^8]. To this end, we will employ the notation of @abramowitz_handbook_1964 [chapter 18].

The elliptic function $\wp\left( z;g_2,g_3\right)$ is a doubly-periodic complex-valued function of a complex variable $z$ defined in terms of two complex parameters $g_2$ and $g_3$, called *invariants*. The complex primitive half-periods $\omega$ and $\omega^\prime$ of $\wp$ can be related to the invariants via formulæ involving elliptic integrals and the roots $e_1$, $e_2$ and $e_3$ of the characteristic cubic equation $$4t^3-g_2t-g_3 = 0 \label{eq:cubic_wp}$$ [e.g., see @abramowitz_handbook_1964 §18.9]. The sign of the *modular discriminant* $$\Delta = g_2^3-27g_3^2$$ determines the nature of the roots $e_1$, $e_2$ and $e_3$. In the case of the Stark problem, the invariants are by definition real (see eqs. –), and thus the $(\omega,\omega^\prime)$ pairs can be chosen as (real, imaginary) or complex conjugate (depending on the sign of $\Delta$). It is known from the theory of elliptic functions that there actually exist infinite pairs of fundamental half-periods for $\wp$, related to each other via integral linear combinations with unitary determinant [@hancock_lectures_1910 §79]. We can then always introduce two new half-periods $\omega_R$ (the *real* period) and $\omega_C$ (the *complex* period) such that $\omega_R$ is real and positive, and $\omega_C$ complex with positive imaginary part. 