# Tagged Questions

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### Proving that $\frac{1}{n}\int_{-\infty}^{\sqrt{n}w}k(v/\sqrt{n})\phi(v)dv$ is $O(n^{-1})$

Suppose that $h:\mathbb{R}\to\mathbb{R}$ is infinitely differentiable. Define k(w)=\left\{ \begin{array}{ll} \frac{d}{dw}\left(\frac{h(w)-h(0)}{w}\right)&w\neq 0,\\ ...
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### Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
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### Simplification of a polynomial before Asymptotic series expansion

I am wondering about a very basic point related to "Asymptotic series expansions". There is a function $f(R)$ which must be expanded as $R$ goes to $\infty$. Consider that $f(R)=g(R)*p(R)$ where ...
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### Some conditions to obtain that $\int_1^{x}e^{f(t)}dt\sim_{x \rightarrow +\infty}\frac{\exp(f(x))}{f'(x)}$

Playing with the function $e^{t^2}$ I conjectured the following result : Let $f\in C^2(\Bbb{R},\Bbb{R})$, assume that : $f'(x)\rightarrow_{x \rightarrow +\infty}+\infty$ ...
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### Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
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### Asymptotics of an oscillatory integral with a linear oscillator

I am interested in asymptotic results for $$S(p) = \int_0^1 \frac{y \sqrt{1-y^2}}{(\varepsilon^2-1)y^2+1} \sin(py) dy,$$ i.e. a result that is valid as $p\rightarrow\infty$. The parameter ...
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### Laplace's Method Integration

Consider the integral $$I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt$$ Use Laplace's Method to show that I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
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### Integration by expansion

Consider the integral $$I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt$$ show that $$I(x)= \frac{2x}{\pi} +O(x^{3})$$ as ...
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### Laplace's Method (Integration)

Consider the integral $$I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt$$ Use Laplace's Method to show that I(x) \sim ...
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### Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
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### Expansion of Integration

Consider the integral $$I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt$$ show that I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
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### Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi$

Is there a closed form for $|r|<1$ and $\ell>0$ integer? The solution for the special cases $\ell=2$ and $4$ would also be interesting if the general case is not available. Integrating ...
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### Integration by parts in vector calculus

I have an axi-symmetric integral (the domain and all functions are axi-symmetric) in cylindrical coordinates which needs to be integrated by parts for use in a finite element code. The integral is ...
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The function $\Phi:(0,\infty) \mapsto \mathbb{R}$ is defined as follows. We put $\Phi(x):=1$ if $x \ge 1$. Let the function $\Phi$ satisfy $$\Phi(x)=\int_0^x \Phi\left(\frac t {1-t}\right) \frac {dt} ... 1answer 76 views ### How to find asymptotics of integrand? Let  f \in C ([0, \infty))  be s. t.$$f(x) \int_0^x f(t)^2 dt \to 1, x \to \infty.$$How to prove that f(x) \sim \left( \frac 1 {3x} \right)^{1/3}  as x \to \infty? 1answer 84 views ### order of magnitude analysis Could anyone explain how to keep track of the error terms when solving an integral approximately? For example consider to evaluate the integral \int_0^{\pi/2}\frac{\cos^2xdx}{x^2+\epsilon^2} as ... 2answers 53 views ### Prove the following: Product of Roots 1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} .... diverges well I don't really know if it does but my gut tells me it does: I can take the log of this product to ... 3answers 282 views ### Asymptotic for the integral involving exponential The integrand seems extremely easy:$$I_n=\int_0^1\exp(x^n)dx$$I want to determine the asymptotic behavior of I_n as n\to\infty. It's not hard to show that \lim_{n\to\infty}I_n=1 follows from ... 1answer 106 views ### Integral of smooth function Another prelim problem: Suppose that f(x,y) is a smooth function defined on \mathbf{R}^2. Prove that$$ \int_{x^2+4y^2\leq r^2}f(x,y)\,dx\,dy = ar^2+br^4+O(r^5)  Express $a$, and $b$ in terms ...
Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. What is the leading order in $\lambda$ as $\lambda\rightarrow 0$ of ...