Bessel functions

Question

The definition of the normalized Bessel functions is commonly understood to be: $$j_{\nu}(x)=\Gamma(\nu+1)\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!\Gamma(n+\nu+1)}(\frac{x}{2})^{2n}.$$

The question is why we have to choose $\nu>-\frac{1}{2}$.

Answer 1

The Bessel function has asymptotics

$J_{\alpha}(x)\sim \frac{1}{\Gamma(1+\alpha)} (x/2)^{\alpha}$ for $x$ close to zero, my guess is for $L^2$ integrability, you need $\alpha>-1/2$.