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At equal argument and large integer order, the Bessel function of the first kind has phase $n(\sin t-t)=-n{t}^3/6+\cdots$ near the endpoint. The cubic stationary endpoint yields the displayed positive constant. It is also the zero-argument value of the Airy function in the uniform Bessel turning-point approximation.

The Bessel turning-point asymptotic comes from a cubic stationary endpoint, not an ordinary quadratic stationary point. Near $\vartheta =0$,Thus the contributing width is $\vartheta =O(n^{-1/3})$. On writing $\vartheta =(6/n{)}^{1/3}s$, the leading integral isThe oscillatory integral is understood with a vanishing damping factor. Substitution $u=s^3$ and the Gamma function Fourier integral giveThereforeThe equality uses the Gamma reflection formula. Contributions away from the degenerate endpoint are smaller. The $n^{-1/3}$ scale explains why the preceding $n^{-1/2}$ formula cannot be extended directly to zero angle.