Can the hypergeometric function be extended analytically to the complex plane in the interval [1,$\infty$ )?

Question

Just a thought. The hypergeometric function, which can be written as:

$$F(a,b,c \space;z) = \frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt$$

is obviously continuous on the unit disc. I was wondering if it could be extended to the positive real line?

Answer 1

The contour integration representation (Barnes integral) allows one to analytically continue the hypergeometric function to the entire complex plane. See Roelof Koekoek's lecture supplement Barnes integral representation. -- Random Variable

Answer 2

Use the Transformation formulae $_2F_1(a,b;c;z)=(1-z)^{-b}{}_2F_1\left(b,c-a;c;\dfrac{z}{z-1}\right)$ or $_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1\left(a,c-b;c;\dfrac{z}{z-1}\right)$ in e.g. http://en.wikipedia.org/wiki/Hypergeometric_function#Transformation_formulas