= Large-argument asymptotic expansion of the Bessel function of the first kind
For fixed $\nu$, put
$$
\chi=z-\frac{\pi\nu}{2}-\frac\pi4,
\qquad
a_k(\nu)=\frac{\prod_{j=1}^k\left(4\nu^2-(2j-1)^2\right)}{k!8^k},
\qquad a_0=1.
$$
Then
$$
J_\nu(z)\sim\sqrt{\frac2{\pi z}}
\left[
\cos\chi\sum_{m=0}^{\infty}\frac{(-1)^ma_{2m}(\nu)}{z^{2m}}
-\sin\chi\sum_{m=0}^{\infty}\frac{(-1)^ma_{2m+1}(\nu)}{z^{2m+1}}
\right]
$$
as $|z|\to\infty$, uniformly for $|\arg z|\leq\pi-\delta$. In particular,
$$
J_\nu(z)\sim\sqrt{\frac2{\pi z}}
\left[
\cos\chi-\frac{4\nu^2-1}{8z}\sin\chi
-\frac{(4\nu^2-1)(4\nu^2-9)}{2!(8z)^2}\cos\chi+\cdots
\right].
$$
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