A truncation error bound for branched continued fractions of the special form on subsets of angular domains

Truncation error bounds for branched continued fractions of the special form are established. These fractions can be obtained by fixing the values of variables in branched continued fractions with independent variables, which is an effective tool for approximating complex functions of two variables. The main result is a two-dimensional analog of the theorem considered in [SCIAM J. Numer. Anal. 1983, 20 (3), 1187$-$1197] for van Vleck's continued fractions. For its proving, the $\mathcal{C}$-figure convergence and estimates of the difference between approximants of fractions in an angular domain are significantly used. In comparison with the previously established results, the elements of a branched continued fraction of the special form can tend to zero at a certain rate. An example of the effectiveness of using a two-dimensional analog of van Vleck's theorem is considered.