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Is there a constant $C$ which is independent of real numbers $a,b,N$, such that

$$\left| {\int_{-N}^N \dfrac{e^{i(ax^2+bx)}-1}{x}dx} \right| \le C?$$

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    After integrating by parts and playing around a little bit, you can easily get an estimate like this provided you make an assumption like $|a| > a_0$, $|b| > b_0$. If you want an estimate that makes *no* such assumption, then I'm not sure there is any reason to expect such an estimate to exist.2012-08-03
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    After some transformation, I need to prove that $$\int_0^N {{e^{ic{x^2}}}\frac{{\sin x}}{x}} dx$$is bounded. But I am stuck there. I am sure that this problem is right since it is an exercise in my book.2012-08-04

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