Suppose I have an $n$-order Bessel function: $J_{n}\left(x\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(nt-x\sin t\right)dt$
I can easily calculate the asymptotics for $x\gg n$. To do this, I will use the stationary phase method. But what should I do if $n\gg x$ ? How to calculate the asymptotics in this case? My intuition tells me that it is necessary to integrate in parts in some way