The two-element arithmetic-geometric mean algorithm, which constitutes an example of approximating a multivalued transcendental function of one variable by means of an algebraic one, was introduced into consideration by C. F. Gauss.

The W. Borchardt algorithm, which is a generalization of the arithmetic-geometric mean algorithm, was first introduced by Borchardt and then studied by I. Hettner.In these works, the Borchardt mean was studied for valid positive initial arguments. The study of the Borchardt mean from complex initial elements is devoted to the work of G. Genpert. The proof of convergence of the Borchardt algorithm is carried out by Geppert on the basis of geometric considerations.In the present paper we give an analytical proof of convergence of the Borchardt algorithm and consider cases of degeneracy of the algorithm.