Chapter 4

Find the asymptotic expansion as \(\lambda \rightarrow \infty\) of the following real integrals using the method of stationary phase: (i) \(\int_{0}^{1} \cos \lambda\left(t^{3}-t\right) d t\) (ii) \(\int_{-\pi / 2}^{\pi / 2} \cos (n t-\lambda \cos t) \mathrm{d} t, n\) an integer; (iii) \(\int_{-\infty}^{\infty} \mathrm{e}^{i 2 t^{2}}\left(1+t^{2}\right)^{-1} \mathrm{~d} t\).

Fresnel's integrals are defined by $$ C(x)+\mathrm{iS}(x)=\int_{0}^{\pi} \mathrm{e}^{\mathrm{i} t^{2}} \mathrm{~d} t $$ If $$ \begin{aligned} &C(x)=4\left(\frac{\pi}{2}\right)^{4}-P(x) \cos x^{2}+Q(x) \sin x^{2} \\ &S(x)=\frac{1}{2}\left(\frac{\pi}{2}\right)^{\frac{1}{2}}-P(x) \sin x^{2}-Q(x) \cos x^{2} \end{aligned} $$ show that $$ P(x) \sim \frac{1}{4 x^{3}}, \quad Q(x) \sim \frac{1}{2 x} \text { as } x \rightarrow \infty . $$