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I want to find the asymptotic expansion of $$\int_0^{\pi/2}e^{-z\sin^2(t)} \, dt$$

As I need to find the entire asymptotic expansion (and not just the first term or two), it suggests I don't want to use Laplace's method (at least not on its own).

Which method would be most appropriate to use in this situation?

Additionally, as a rough idea, when is it best to use each technique?

user112495
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1 Answers1

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You can try with the fact that

$$ \int_{0}^{\pi/2}{\rm d}t~e^{-z\sin^2t}=\frac{\pi}{2}e^{-z/2}I_0\left(\frac{z}{2}\right) \tag{1} $$

with $I_0$ being the modified Bessel function of of the first kind

From this, obtaining an asymptotic behavior is straightforward, for instance, for large $z$ we have

$$ I_\alpha(z) = \frac{e^z}{\sqrt{2\pi z}}\left( 1 - \frac{4\alpha^2 - 1}{8z} + \frac{(4\alpha^2 - 1)(4\alpha^2 - 9)}{2!(8z)^2} - \cdots\right) \tag{2} $$

So Eq. (1) becomes

$$ \int_{0}^{\pi/2}{\rm d}t~ e^{-z\sin^2t} \approx \sqrt{\frac{\pi}{4z}} \left(1 + \frac{1}{4z} + \frac{9}{32z^2} + \cdots \right) $$

caverac
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