ABSTRACT
For the principal branch with 121 < 1 this result is verifiable from the definition (9.03): the M-test shows that this series converges uniformly in any bounded region of the complex a, b, c space. The extension to lzl2 1 and other branches is immediately achieved by means of Theorem 3.2; any point within the unit disk, other than the origin, may be taken as z0 in Condition (iv) of this theorem. The points z = 0, 1, and a are excluded in the statement of the final result, because F(a, b; c; z) may not exist there.' 9.3 Many well-known functions are expressible in the notation of the hypergeometric function. For example, the principal branch of (1 -2)-" is also the principal branch of F(a, 1 ; 1 ;z). Other examples are stated in Exercises 9.1, 9.2, and 10.1 below.
