On the convergence of continued fractions with complex elements

Up to the present time few theorems of a general character for the conivergence of continued fractions with complex elements have been obtained, and these few are of very recent date. In the first section of this paper such theorems upon the subject as are known to the writer are brought together for the purpose of indicating the present state of our knowledge, and the scope of the paper is also explained. Some new criteria for convergence are then deduced in the succeeding sections. The results obtained are summed up in theorems 1-10, which may be read independently of the rest of the paper. The demonstration of these theorems is based upon certain equations, Nos. 3-8, 11, and 12, which seem to be new and of a fundamental character.