Solution
= Solution
Take $n=1$ in part (c). The original equation $w_{xx}+w=0$ has the solution $\sin x=\sqrt{1-z^2}$. The solution $w_2$ is normalized by $w_2/(1-z)^{1/2}\to1$ as $z\to1$, so
$$
w_2(z)=\frac1{\sqrt2}\sqrt{1-z^2}
=(1-z)^{1/2}\sqrt{\frac{1+z}{2}}.
$$
Comparing this with the formula for $w_2$ and using $u=(1-z)/2$ gives
$$
\boxed{F\left(-\frac12,\frac32;\frac32;u\right)
=\sqrt{1-u}.}
$$
This also agrees with the general identity $F(A,C;C;u)=(1-u)^{-A}$.