The Laplace’s Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g”(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where $g(x_0)$ is the maximum of $g$ and $g”(x_0)$ is the second derivative at that point. (At that point $g”(x_0)<0$. Also $x_0$ is in $(a,b)$.) Inspired in this […]

I’m still having a little trouble applying Laplace’s method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like: $$\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$$ for $x>0$, as $x\rightarrow\infty$.

Let \begin{align} f(t,k,p)= \frac{ \int_0^\infty \cos(x t) e^{-x^k}dx}{\int_0^\infty \cos(x t) e^{-x^p}dx}, \end{align} My question: How to find the following limit of the function $f(t,k,p)$ \begin{align} \lim_{t \to \infty} f(t,k,p), \end{align} for any $p>0$ and $k>0$. What is known Some facts about the function Note that $\int_0^\infty \cos(x t) e^{-x^k}dx$ is a fourierier transform of $e^{-{|x|^k}}$. […]

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