  Here we construct the conformal mappings with the help of continuous
fractions approximations. These approximations converge to the algebraic roots
$\sqrt[N]{z}$ for $N \in \mathbb{N}$ and $z$ from the right half-plane of the
complex plane. We estimate both the convergence rate and the compact set of
convergence. Also we give the examples that illustrate the introduced technique
of a conformal mapping construction.
