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During my research I ran into the following type of an oscillatory integral, for some values of nonzero reals $a,b$:

$f(R):=\int_{0}^{R} e^{2 \pi i (ar^2 + br)} dr$

and I am interested in finding a good estimate for it when $R>0$ is large, in terms of $R$ and the two parameters $a,b$. I expect $\frac{1}{R}f(R)$ to converge as $R \to \infty$ uniformly in $b$, but a more quantitative estimate will be helpful. I guess this is classical but since I know very little in this area I'm not sure where to find such results.

EDIT: Thanks for your help. In the end I just used a version of van der Corput's lemma, but the answers offered here were still interesting to read.

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    Any knowledge about $a$ and $b$?2012-07-09
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    If $a\ne0$ is real or $\Im a>0$ the integral from zero to $\infty$ converges.2012-07-09
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    But the asymptotics might depend on the sign of the parameters... So it would be helpful to know more about the signs to give the appropriate asymptotic expression.2012-07-09
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    @Fabian: any two real numbers, both nonzero.2012-07-09

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