How Does the Method of Stationary Phase Apply to Asymptotic Expansions?

[math2011]

Homework Statement

Find asymptotic expansions for the following integral, as $x \to \infty$, using the method of stationary phase.

$$\int^1_0 e^{ixt^p} dt, p > 1$$

Homework Equations

The Attempt at a Solution

1. let the phase be $t^2$. is this correct?
2. $\frac{d}{dt} t^2 = 2t = 0$, so phase stationary is $0$.
3. expand phase about stationary point $t=0$ and i get $0$. something is wrong, should i be doing this?

 

[Adrmay]

4. is the phase t^p?5. \frac{d}{dt} t^p = pt^(p-1), so phase stationary is 0. 6. expand phase about stationary point t=0 and i get 0. 7. \int^1_0 e^{ixt^p} dt = \int^1_0 1 + ixpt^p + \frac{(ixp)^2}{2!} t^{2p} + o(x^{-3})dt 8. \int^1_0 e^{ixt^p} dt = \int^1_0 1 + ixpt^p + o(x^{-3})dt 9. \int^1_0 e^{ixt^p} dt = 1 + \frac{(ixp)^2}{2!} \int^1_0 t^{2p}dt + o(x^{-3})10. \int^1_0 e^{ixt^p} dt = 1 + \frac{(ixp)^2}{2!} \frac{1}{2p+1} + o(x^{-3})