Question 10.33: Show that F(1/2, 1/2; 3/2;z²) = sin^-1 z/z .

\begin{aligned} F\left(1 / 2,1 / 2 ; 3 / 2 ; z^{2}\right)= & 1+\frac{(1 / 2)(1 / 2)}{1 \cdot(3 / 2)} z^{2}+\frac{(1 / 2)(3 / 2)(1 / 2)(3 / 2)}{1 \cdot 2 \cdot(3 / 2)(5 / 2)} z^{4} \\ & +\frac{(1 / 2)(3 / 2)(5 / 2)(1 / 2)(3 / 2)(5 / 2)}{1 \cdot 2 \cdot 3 \cdot(3 / 2)(5 / 2)(7 / 2)} z^{6}+\cdots \\ = & 1+\frac{1}{2} \frac{z^{2}}{3}+\frac{1 \cdot 3}{2 \cdot 4} \frac{z^{4}}{5}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^{6}}{7}+\cdots=\frac{\sin ^{-1} z}{z} \end{aligned}