The hypergeometric function $F(a, b ; c ; z)$ is defined by

$$F(a, b ; c ; z)=K \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

where $|\arg (1-t z)|<\pi$ and $K$ is a constant determined by the condition $F(a, b ; c ; 0)=1$.

(i) Express $K$ in terms of Gamma functions.

(ii) By considering the $n$th derivative $F^{(n)}(a, b ; c ; 0)$, show that $F(a, b ; c ; z)=F(b, a ; c ; z)$.