Gauss's continued fraction

In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions, as well as some of the more complicated transcendental functions. Lambert published several examples of continued fractions in this form in 1768, and both Euler and Lagrange investigated similar constructions, but it was Carl Friedrich Gauss who utilized the algebra described in the next section to deduce the general form of this continued fraction, in 1813. Although Gauss gave the form of this continued fraction, he did not give a proof of its convergence properties. Bernhard Riemann and L.W. Thomé obtained partial results, but the final word on the region in which this continued fraction converges was not given until 1901, by Edward Burr Van Vleck. Let be a sequence of analytic functions so that for all , where each is a constant. Then Setting So Repeating this ad infinitum produces the continued fraction expression In Gauss's continued fraction, the functions are hypergeometric functions of the form , , and , and the equations arise as identities between functions where the parameters differ by integer amounts. These identities can be proven in several ways, for example by expanding out the series and comparing coefficients, or by taking the derivative in several ways and eliminating it from the equations generated. The simplest case involves Starting with the identity we may take giving or This expansion converges to the meromorphic function defined by the ratio of the two convergent series (provided, of course, that a is neither zero nor a negative integer). The next case involves for which the two identities are used alternately. Let etc. This gives where , producing or Similarly or Since , setting a to 0 and replacing b + 1 with b in the first continued fraction gives a simplified special case: The final case involves Again, two identities are used alternately.