Bessel turning-point asymptotic ($J_n(n)\sim 2^{1/3}{n}^{-1/3}/[3^{2/3}\Gamma (2/3)]$)

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.