formula $$\begin{aligned}
{U}_m=\frac{\epsilon}{ 2\pi^g R} \mathrm{Cofactor}\left(  \left.  \frac{ \partial( \theta[\varepsilon_1]({v}),\ldots,
            \theta[\varepsilon_g]({v}))  }
        {\partial (v_1,\ldots, v_g)}\right|_{{v}=0}
 \right)
\left( \begin{array}{c}  s_{m-1}^{2} \sqrt[4]{\chi_1}\\ \vdots
\\s_{m-1}^{2g} \sqrt[4]{ \chi_g}\end{array}  \right)\end{aligned}$$ Here $s_k^i$ -order $k$ symmetric function of elements $\{e_{2}, \ldots e_{2g}  \}  /  \{ e_{2i} \}$ and $$\chi_{i} = \frac{\psi(e_{2i})}{\phi'(e_{2i})} , \quad i=1,\ldots, g$$ $s_k^i, \chi_i$ are expressible in $\theta$-constants via Thomae formulae [@tho870].

Typical answer of multi-gap integration includes $\theta$-function $  \theta({U}x +{V}t+{W};\tau )$ where winding vectors ${U}, {V}$ expressed in terms of complete holomorphic integrals and constant ${W}$ defined by initial data. Rosenhain formulae and their generalization, express $U,V$ in terms of $\theta$-constants and parameters the equation defining $\mathcal{C}$. In this way the problem of [*effectivization of finite gap solutions*]{} [@dub81] can be solved in that way at least for hyperelliptic curves.

Other application of the Rosenhain formula (\[RosenhainFormula\]) presented in [@be994] where two-gap Lamé and Treibich-Verdier potentials were obtained by the reduction to elliptic functions of general Its-Matveev representation [@im975] of finite-gap potential to the Schrödinger equation in terms of multi-variable $\theta$-functions.

Another application relevant to a computer algebra problem. In the case when, say in Maple, periods of holomorphic differentials are computed them periods of second kind differentials can be obtained 