An analytic function is generally given either directly as a power-series, or at anyrate in some form which can readily be converted into a power-series. The power-series is not an altogether satisfactory method of representing the function, on account of the failure of its convergence outside its circle of convergence; in this respect it is decidedly inferior to the method of representing the function by a continued fraction, as this latter expression in many cases converges over the whole plane, except on certain curves.