i.e. $e_i\neq e_j$.

Representations of complete elliptic integrals of first and second kind in terms of Jacobi $\theta$-constants are classically known. In particular, if elliptic curve is given in Legendre form[^1] $$y^2=(1-x^2)(1- k^2x^2)$$ where $k$ is Jacobian modulus, then complete elliptic integrals of the first kind $K=K(k)$ is represented as $$\begin{aligned}
    K=  \int_{0}^1 \frac{\mathrm{d} x}{ \sqrt{ (1-x^2)(1-k^2 x^2) }}= \frac{\pi}{2} \vartheta_3^2(0;\tau), \label{FirstKind}
 \end{aligned}$$ and $\vartheta_3=\vartheta_3(0|\tau)$ and $\tau=\imath \frac{K'}{K}$, $K'=K(k'),  k^2+{k'}^2=1$.

Further, for elliptic curve realized as Weierstrass cubic $$y^2= 4 x^3-g_2 x - g_3=  4 (x-e_1)(x-e_2)(x-e_3) \label{WCubic}$$ recall standard notations for periods of first and second kind elliptic integrals $$\begin{aligned}
  \begin{split}
  2\omega &= \oint_{\mathfrak{a}} \frac{\mathrm{d}x}{y}, \quad  2\eta = -\oint_{\mathfrak{a}} \frac{x\mathrm{d}x}{y}\\
  2\omega' &= \oint_{\mathfrak{b}} \frac{\mathrm{d}x}{y}, \quad  2\eta' = -\oint_{\mathfrak{b}} \frac{x\mathrm{d}x}{y}
\end{split} \hskip1.5cm    \tau = \frac{\omega'}{\omega}\end{aligned}$$ and Legendre relation for them $$\quad\omega\eta'-\eta\omega' = - \frac{\imath \pi}{2}  \label{LegendreRel}$$ Then the following Weierstrass relation is valid $$\begin{aligned}
 \eta=-  \frac{1}{12\omega} \left(\frac{\vartheta_2''(0)}{\vartheta_2(0)}
+ \frac{\vartheta_3''(0)}{\vartheta_3(0)}+ \frac{\vartheta_4''(0)}{\vartheta_4(0)}\right)
\label{SecondKind}\end{aligned}$$ In this section we are discussing generalization of these relations to higher genera hyperelliptic curves realized as in (\[HCurve\]).

In this subsection we reproduce H.Baker [@bak907] notations. Let $\mathcal{C}$ be genus $g$ non-degenerate hyperelliptic curve realised as double cover of Riemann sphere, $$\begin{aligned}
y^2= 4 \prod_{j=1}^{2g+1}(x-e_j)\equiv 4x^{2g+1}+ \sum_{i=0}\lambda_i x^i,\quad e_i\neq 