Let $w_{1}(z)$ and $w_{2}(z)$ be any two linearly independent branches of the $P$-function

$$\left\{\begin{array}{cccc} 0 & \infty & 1 & \\ \alpha & \beta & \gamma & z \\ \alpha^{\prime} & \beta^{\prime} & \gamma^{\prime} & \end{array}\right\}$$

where $\alpha+\alpha^{\prime}+\beta+\beta^{\prime}+\gamma+\gamma^{\prime}=1$, and let $W(z)$ be the Wronskian of $w_{1}(z)$ and $w_{2}(z)$.

(i) How is $W(z)$ related to the Wronskian of the principal branches of the $P$-function at $z=0$ ?

(ii) Show that $z^{-\alpha-\alpha^{\prime}+1}(1-z)^{-\gamma-\gamma^{\prime}+1} W(z)$ is an entire function.

(iii) Given that $z^{\beta+\beta^{\prime}+1} W(z)$ is bounded as $z \rightarrow \infty$, show that

$$W(z)=A z^{\alpha+\alpha^{\prime}-1}(1-z)^{\gamma+\gamma^{\prime}-1}$$

where $A$ is a non-zero constant.