Gauss's Continued Fraction - Convergence Properties

Convergence Properties

In this section, the cases where one or more of the parameters is a negative integer are excluded, since in these cases either the hypergeometric series are undefined or that they are polynomials so the continued fraction terminates. Other trivial exceptions are excluded as well.

In the cases and, the series converge everywhere so the fraction on the left hand side is a meromorphic function. The continued fractions on the right hand side will converge uniformly on any closed and bounded set that contains no poles of this function.

In the case, the radius of convergence of the series is 1 and the fraction on the left hand side is a meromorphic function within this circle. The continued fractions on the right hand side will converge to the function everywhere inside this circle.

Outside the circle, the continued fraction represents the analytic continuation of the function to the complex plane with the positive real axis, from +1 to the point at infinity removed. In most cases +1 is a branch point and the line from +1 to positive infinity is a branch cut for this function. The continued fraction converges to a meromorphic function on this domain, and it converges uniformly on any closed and bounded subset of this domain that does not contain any poles.