by Rosenhain formula (\[RosenhainFormula\]) and its generalization.

Sigma-functions and the problem of differentiatiion of Abelian functions.
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Consider the curve $$\label{F-1}
V_\lambda = \left\{(x,y)\in\mathbb{C}^2\, :  y^2 = \mathcal{C}(x;\lambda) = x^{2g+1}+\sum_{k=2}^{2g+1}
\lambda_{2 k} x^{2g - k + 1} \right\}$$ where $g\geqslant 1$ and $\lambda=(\lambda_4,\ldots,\lambda_{4g+2})\in \mathbb{C}^{2g}$ are the parameters. Set $\mathcal{D} = \{ \lambda\in \mathbb{C}^{2g}\,: \mathcal{C}(x;\lambda)\; \text{has multiple roots}\}$ and $\mathcal{B} = \mathbb{C}^{2g}\setminus\mathcal{D}$. For any $\lambda\in \mathcal{B}$ we obtain the affine part of a smooth projective hyperelliptic curve $\overline{V}_\lambda$ of genus $g$ and the Jacobian variety $Jac(\overline{V}_\lambda) = \mathbb{C}^{g}/\Gamma_g$, where $\Gamma_g \subset \mathbb{C}^{g}$ is a lattice of rank $2g$ generated by the periods of the holomorphic differential on cycles of the curve $V_\lambda$.

In the general case, an *Abelian function* is a meromorphic function on a complex Abelian torus $T^g=\mathbb{C}^g\!/\Gamma$, where $\Gamma\subset\mathbb{C}^g$ is a lattice of rank $2g$. In other words, a meromorphic function $f$ on $\mathbb{C}^g$ is Abelian iff $f(u)=f(u+\omega)$ for all $u=(u_1,\ldots,u_g)\in\mathbb{C}^{g}$ and $\omega\in\Gamma$. Abelian functions on $T^g$ form a field $\mathcal{F} = \mathcal{F}_g$ such that:

\(1) let $f\in\mathcal{F}$, then $\partial_{u_i} f\in \mathcal{F}$, $i=1,\dots,g$;

\(2) let $f_1,\dots,f_{g+1}$ be any nonconstant functions from $\mathcal{F}$, then there exists a polynomial $P$ such that $P(f_1,\dots, f_{g+1})(u)=0$ for all $u\in T^g$;

\(3) let $f\in\mathcal{F}$ 