By definition of Bessel function,

$J_{1/2} (x) = \displaystyle \sum^\infty _ {m=0} \dfrac {(-1)^m (x/2)^{2m+n}}{m! |\overline {n+m+1}}$

$\displaystyle \therefore J_{1/2}(x) = \sum^{\infty}_{m=0} \dfrac {(-1)^m}{m! |\overline {m+3/2}} \times \dfrac {x^{2m+1/2}}{2^{2m+1/2}}$

$= \dfrac {x^{1/2}}{0!\cdot 2^{1/2}|\overline{3/2}} - \dfrac {x^{5/2}}{1!\cdot 2^{5/2}| \overline {5/2}} + \dfrac {x^{9/2}}{2!\cdot 2^{2/9} |\overline {7/2}} \ \cdots \to (1)$

We know, 

$|\overline {n} = (n-1) |\overline {n-1}$

$\therefore \Gamma\left ( \dfrac {7}{2} \right )= \dfrac {5}{2} \Gamma \left ( \dfrac {5}{2} \right ) = \dfrac {5}{2} \cdot \dfrac {3}{2} \cdot \Gamma \left ( \dfrac {3}{2} \right )= \dfrac {5}{2} \cdot \dfrac {3}{2}\cdot \dfrac {1}{2} \cdot \Gamma \left ( \dfrac {1}{2} \right ) = \dfrac {5}{2} \cdot \dfrac {3}{2} \cdot \dfrac {1}{2} \sqrt{\pi}$

Similarly,

$\Gamma \left ( \dfrac {5}{2} \right )= \dfrac {3}{2} \cdot \dfrac {1}{2} \sqrt{\pi} , \ \Gamma \left ( \dfrac {3}{2} \right ) = \dfrac {1}{2} \sqrt {\pi}$

∴ From (1),

$j_{1/2}(x) = \left [ \dfrac {x^{1/2}}{2^{1/2}} \times \dfrac {2}{\sqrt{\pi}} - \dfrac {x^{5/2}}{2^{5/2}} \times \dfrac {2}{3} \times \dfrac {2}{\sqrt{\pi}}+ \dfrac{x^{9/2}}{2\times 2^{9/2}} \times \dfrac {2}{5} \times \dfrac {2}{3} \times \dfrac {2}{\sqrt{\pi}} - \cdots \right ] \times \dfrac {x^{1/2}}{x^{1/2}}$

$= \dfrac {1}{\sqrt{x}} \left [ \dfrac {2^{1/2}x}{\sqrt{x}} - \dfrac {x^3}{2^{1/2}\times 3\sqrt{\pi}} + \dfrac {x^5}{2^{5/2} \times 15 \sqrt{\pi}} - \cdots \right ]$

$= \dfrac {1}{\sqrt{x}} \times \dfrac {2^{1/2}}{\sqrt{\pi}} \left [ \dfrac {x}{1} - \dfrac {x^3}{2^1 \times 3} + \dfrac {x^5}{2^3 \times 15} - \cdots \right ]$

$= \sqrt{\dfrac {2}{\pi x}} \left ( x - \dfrac {x^3}{3!} + \dfrac {x^5}{5!} - \cdots \right )$

$=\sqrt{ \dfrac {2}{\pi x}} \sin x$