If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that
$$
\Big|\int\_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\lambda
$$
as $\lambda\rightarrow\infty$.


$\textbf{My question}$ is, whether one can determine the rate of decay, in terms of $\lambda$, of the oscillatory integral
$$
\Big|\int\_0^1 \frac{1}{\sqrt{x}}\, e^{i \lambda x}\, dx\Big|?
$$
Since $\frac{1}{\sqrt{x}}\in L^1((0,1))$, it follows by the Riemann-Lebesgue Lemma that $\int\_0^1 \frac{1}{\sqrt{x}}\, e^{i \lambda x}\, dx$ vanishes as $\lambda\rightarrow\infty$. However, Riemann-Lebesgue Lemma doesn't say anything about the rate of decay, hence my question.


Thanks for reading!



Estimate for a simple oscillatory integral
By the substitution $tx=u$, the integral in question is
$$\int\_0^1\frac{e^{itx}}{\sqrt x}\,dx=\frac1{\sqrt t}\,\int\_0^t\frac{e^{iu}}{\sqrt u}\,du
\sim\frac1{\sqrt t}\,\int\_0^\infty\frac{e^{iu}}{\sqrt u}\,du
=(1+i) \sqrt{\frac{\pi }{2}}\frac1{\sqrt t}$$
as $t\to\infty$.


