The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$

(i) if we set $v=-1/2$ then we have $\Gamma{(1+v+n)}=\Gamma{(\frac{1}{2}+n)}$. Use suitable properties of the gamma function to find an expression for $\Gamma{(\frac{1}{2}+n)}$ in terms of $\Gamma{(\frac{1}{2})}$.

(ii) using the fact that $\Gamma{(\frac{1}{2})}=\sqrt{\pi}$, show that $$J_{-1/2}(x)=\sqrt{\frac{2}{\pi x}} \cos(x)$$

I think both is just using mathematical manipulation. I don't think I need to use the $J_v$ equation but I am not sure.

Any help would be awesome.


About the first request, it is the Legendre duplication formula (you can find a proof here) and we get $$\frac{1}{\Gamma\left(n+1/2\right)}=\frac{n!4^{n}}{\left(2n\right)!\Gamma\left(1/2\right)}\tag{1} .$$ About the second request, we have, using $(1)$ $$J_{-1/2}\left(x\right)=\left(\frac{x}{2}\right)^{-1/2}\sum_{n\geq0}\frac{\left(-1\right)^{n}}{n!\Gamma\left(n+1/2\right)}\left(\frac{x}{2}\right)^{2n} $$ $$=\left(\frac{x}{2}\right)^{-1/2}\frac{1}{\sqrt{\pi}}\sum_{n\geq0}\frac{n!\left(-1\right)^{n}4^{n}}{n!\left(2n\right)!}\left(\frac{x}{2}\right)^{2n} $$ $$=\left(\frac{x}{2}\right)^{-1/2}\frac{1}{\sqrt{\pi}}\sum_{n\geq0}\frac{\left(-1\right)^{n}}{\left(2n\right)!}x^{2n} $$ $$=\sqrt{\frac{2}{x\pi}}\cos\left(x\right) $$ from the Taylor series of the cosine function.


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