CHAPTER VI - THE GAUSSIAN HYPERGEOMETRIC FUNCTION 2F1

Abstract

This chapter discusses the Gaussian hypergeometric function. Gauss proved that between the 2F1 and any two functions contiguous to it, there exists a linear relation with coefficients which are linear in z. There are 15 relations of this kind. Only four of the 15 are really independent, as all others may be obtained by elimination and use of the fact that the 2F1 is symmetric in a and b. The integral equation defines a single valued analytic function of z in the domain |arg (1-z)|< π, and so serves for the analytic continuation of the 2F1 hypergeometric series into this domain. It is convenient to denote the analytic continuation of the 2F1 series by 2F1, and this means the principal branch of the analytic function generated by the hypergeometric series.