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Asymptotic evaluation of $\int_0^{\pi/4}\cos(x t^2)\tan^2(t)dt$

In Bender-Orszag's Advanced Mathematical Methods for Scientists and Engineers on page 313 we encounter the following integral $$I(x)=\int_0^{\pi/4}\cos(x t^2)\tan^2(t)dt$$ and it is asked to ...
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Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$

Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \...
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Evaluate $\lim_{r\to\infty}\frac{\int_0^{\pi/2}x^{r-1}\cos x\,\mathrm dx}{\int_0^{\pi/2}x^r\cos x\,\mathrm dx}$

How I can evaluate $$\lim_{r\to\infty}\frac{\int_0^{\pi/2}x^{r-1}\cos x\,\mathrm dx}{\int_0^{\pi/2}x^r\cos x\,\mathrm dx}$$ I have tried by replacing $x$ with $y\pi/2$ then the limits would change ...
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Higher order corrections to saddle point approximation

I'd like to ask for hints how to obtain higher order corrections to approximations obtained by the saddle point method. References will be also welcome. Unfortunately what comes up when googling is ...
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Tricky steepest descent applied to an inverse Fourier transform

I answered a question a while ago on solving a PDE, and ended up with the solution in terms of an inverse Fourier transform but left it at that. I'm curious to try and approximate it now, using the ...
160 views

Steepest descent of integrand with a movable saddle?

I want to apply the steepest descent method to the following integration: $$\int_0^\infty e^{-x^2 + i \sqrt{x^2 + 1} \cdot \lambda } dx$$ It has movable saddle so I need to transform it into the ...
210 views

Zeros of Fox-Write Function

This question is stimulate by the previous two question here and here. We are interested in studying the following special case of Fox-Write function \begin{align} \Psi_{1,1} \left[ \begin{array}{...
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Existence of additive transformation of random variables

Suppose we have a random variable $W$ and we want to transform it to a random variable $V$ by using additive transformation, as follows \begin{align} V=U+W \end{align} where $U$ is independent of $W$. ...
Fourier transform of $\exp(-z^k)$: How can one quatify its decay?
Consider the Fourier transform of $\exp(-z^k)$ where $k$ is a positive integer. As the function is analytic, I expect it to have exponential decay at infinity. Is there some known theorem giving a ...