Find this limit $$\lim_{x\to \infty}\sqrt{x}\int_{0}^{\frac{\pi}{4}}e^{x(\cos{t}-1)}\cos{t}dt$$
maybe use $$\cos{t}-1\approx-\dfrac{t^2}{2}$$
But I can't.Thank you
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The limit should be $\sqrt{\pi/2}$. – Mhenni Benghorbal Jan 28 '14 at 06:14
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You can use Laplace's technique . – Mhenni Benghorbal Jan 28 '14 at 06:17
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Indeed, use $\cos t - 1 \sim -t^2/2$, $\cos t \sim 1$, and replace $\int_0^{\pi/4}$ with $\int_0^\infty$. – Antonio Vargas Jan 28 '14 at 06:51
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Your idea of replacing the cosine by its first order development as a Taylor series is interesting; however, I do not know how far we could go to justify it. But, let us admit its validity but let us do it only for the exponential term.
In this case, the integration leads to a quite awful expression i which appear the error functions
$$\frac{1}{2} \sqrt{\frac{\pi }{2}} e^{-\frac{1}{2 x}} \left(\text{erf}\left(\frac{\pi x-4 i}{4 \sqrt{2} \sqrt{x}}\right)+i \text{erfi}\left(\frac{4-i \pi x}{4
\sqrt{2} \sqrt{x}}\right)\right)$$
If now, we take the limit, we find $\sqrt{\frac{\pi }{2}}$.
I must say that I am not very happy of that (even if it leads to the expected result). I shall continue working this problem and, if I get anything better, I shall post it.
Claude Leibovici
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