A very short question here... I'd like to use an asymptotic approximation of a Bessel functions, which I've found given in several places as $J_\nu(z)=\sqrt{\frac{2}{\pi z}}cos(z-\frac{1}{2}\nu \pi -\frac{\pi}{4})$ for $|arg(z)|<\pi$ for large z.
My question is, doesn't this mean that the asymptotic form is valid for all nonnegative arguments z, since $|arg(z)|<\pi$ means $-\pi<arg(z)<\pi$? Am I misunderstanding the references' conditions or is that all there is to it?
I don't know how to derive such an expansion myself, so I don't know where the condition came from. Is it correct that the asymptotic expression is valid for all large z away from the negative real line?
From its expansion, $$J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(% \tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}$$ is an analytic function of $z$, except for a branch point at $z=0$ when $\nu$ is not an integer (see here). In this case, the given condition $\left|\operatorname{Arg}(z)\right|<\pi$ corresponds to the principal branch of $J_\nu(z)$ (cut along the negative real axis).
If $\nu$ is an integer, $J_\nu(-z)=(-1)^\nu J_\nu(z)$, which allows to find the asymptotic expansion for $z<0$.