0
votes
0answers
12 views

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 ...
6
votes
1answer
98 views

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 : $f'(x)\rightarrow_{x \rightarrow +\infty}+\infty$ ...
7
votes
1answer
214 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
6
votes
3answers
143 views

Asymptotic behaviour of the integral of the quadratic mean of the coordinates on the hypercube

I have to compute the limit $\lim_{n\to +\infty}I_n$, where: $$\qquad I_n=\int_{[0,1]^n}\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\,d\mu.$$ I believe that its value is just $\frac{1}{\sqrt{3}}$, since the ...
0
votes
0answers
36 views

Growth rate of integral

My apologies, I have no idea how to make the title more specific without putting the whole question in there. On p. 60 of Montgomery and Vaughan they state \begin{equation} 2\int_e ^x \frac{1 + \log ...
4
votes
1answer
103 views

Asymptotic expansion of $\int\limits_0^{\pi / 2} {e^{ix\cos t}}dt$

Using the method of stationary phase, I was able to obtain the first term of the asymptotic expansion of the following integral, as $x \rightarrow \infty$: $$\int\limits_0^{\pi / 2} {e^{ix\cos t}}dt ...
3
votes
1answer
47 views

Asymptotic approximation to find the Barnes integral

In the following paper and in order to prove the Barnes integral $$\frac{1}{2\pi ...
5
votes
3answers
121 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
4
votes
1answer
70 views

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 ...
5
votes
1answer
112 views

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 ...
0
votes
1answer
51 views

Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
3
votes
2answers
79 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
0
votes
1answer
54 views

Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
3
votes
0answers
64 views

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 ...
0
votes
1answer
51 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
1
vote
1answer
42 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
1
vote
0answers
57 views

How to find Laplace approximation for following integral?

Let's have integral $$ I(x) = \frac{1}{2\pi} \int \limits_{-\pi}^{\pi}e^{xcos(\theta )}d \theta, \quad x \to +\infty . $$ How to use Laplace approximation for this integral and find first two ...
4
votes
1answer
138 views

Stationary phase method for $\int_{-\infty}^{\infty}f(t)\exp(ix(t^3-t))dt$

I am currenty struggling with the integral $\int_{-\infty}^{\infty}f(t)\exp(ix(t^3-t))dt$ where $f(t)$ is smooth and $f\rightarrow 0$ as $t\rightarrow +-\infty$. I want to obtain the leading ...
4
votes
1answer
200 views

Application of Riemann-Lebesgue Lemma

I am considering the integral $\int_a^{\infty}f(t)\cos(\omega t)dt$ and I want to find the asymptotic expansion using the Riemann-Lebesgue Lemma where as $\omega\rightarrow \infty$, $a,\omega$ real ...
0
votes
1answer
35 views

A complex integration problem

The problem is: Let $\gamma$ be the circle of radius $R$ centered at $0$. Let $m$, $n$ be positive integers. Prove that, as $R$ goes to infinity, $\int _\gamma\frac{z^m}{z^n+1}dz=O(R^{m+1-n})$. And ...
2
votes
1answer
74 views

Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+...)$ as $x\rightarrow \infty$ I started by ...
3
votes
1answer
75 views

Application of Laplace Method

I have the integral $I(x)=\int_0^{\pi/2}\exp(-xt^3\cos t)dt$ and I want to ederive the first two terms in an asymptotic expansion as $x\rightarrow \infty$ which should give me ...
6
votes
0answers
81 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
1
vote
1answer
68 views

$\pi(x)$ asymptotic as integral $1/\log t$

From the prime number theorem we know that $\pi(x)\sim x/\log x$, i.e. $\dfrac{\pi(x)\log x}{x}\rightarrow 1$ as $x\rightarrow \infty$. How can we use that to show that ...
4
votes
3answers
230 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
1
vote
0answers
19 views

parametric integral and asymptotic representation

Here is a parametrial integral $$I(a)=\int_0^{\pi}\int_0^{\pi} ...
10
votes
2answers
337 views

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 ...
1
vote
0answers
62 views

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 ...
2
votes
1answer
64 views

asymptotic approximation when $a\to 0^+$ of $I(a):=\int_0^\infty \int_0^{a/x}e^{-x-y}\ dy\ dx.$

I want to find an asymptotic approximation when $a\to 0^+$ for the integral $$I(a):=\int_0^\infty \int_0^{a/x}e^{-x-y}\ dy\ dx.$$ I found the following approximation: $$C_1\, a\, \mathrm{ln}(1/a) ...
2
votes
1answer
80 views

Lommel function

I need to do this integral: $$\int_0^\infty dx\cdot x \sqrt{x^2+1}K_0(ax)$$ where K is the modified Bessel of second kind. I have seen that in Gradhsteyn 7th edition in 6.565.7 says that this ...
2
votes
2answers
178 views

Asymptotic expansion of an integral

I came up with a simpler example which illustrates the technical difficulty I have encountered in my work. Consider an integral depending on parameter $\epsilon$: \begin{equation} ...
1
vote
1answer
301 views

Laplace transformation of a polynomial function involving square root (or approximation of the integral)

Could somebody suggest how to solve this Laplace transform: $$ \int_0^\infty{e^{-at}\over\sqrt{A+Bt+Ct^2}}{\rm d\,}t $$ ? The real coefficients $A,B,C$ are chosen such that the roots of $A+Bt+Ct^2$ ...
5
votes
6answers
265 views

Asymptotic solution to the integral $\int_{-\pi/2}^{\pi/2} (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x$

Recently, I have posted a question on how to find a reduction formula for the trigonometric integral $$\int (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x.$$ This problem seems to be tough, however. When ...
1
vote
1answer
145 views

Calculating expected value of distance in a circle-circle intersection

Consider two circles $c_1$ and $c_2$ both of radius $r$ located in 2-D plane such that the distance between their centers is $r$. Assume a point is randomly and uniformly chosen within their ...
1
vote
0answers
62 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
6
votes
1answer
59 views

Asymptotic behaviour of sum of decreasing definite integrals

I would like to calculate: \begin{equation*}g(K, T) = \displaystyle \sum_{k=1}^{K} \sum_{t = 1}^{T} \int_{0}^{1} \left(1 - z^k\right)^t \, dz. \end{equation*} If no closed form solution exists, I ...
1
vote
0answers
61 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
0
votes
1answer
33 views

Behavior of the following function at x=0 singularity

I am trying to do the following integral: \begin{equation} \int\frac{dx}{x^{2p}(x-1)^{2q}} \end{equation} for positive $2p$ and $2q$. I want to understand how does this function blow up (the $x$ ...
1
vote
0answers
84 views

Solution by of nonlinear equation

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0$$ From the sine-Gordon equation we can easily solve, \begin{equation} \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
4
votes
2answers
236 views

Asymptotic expansion of a function $\frac{4}{\sqrt \pi} \int_0^\infty \frac{x^2}{1 + z^{-1} e^{x^2}}dx$

How to find the asymptotic expansion of the following function for large values of $z$. $$f_{3/2}(z) = \frac{4}{\sqrt \pi} \int_0^\infty \frac{x^2}{1 + z^{-1} e^{x^2}}dx $$ I have to get something ...
3
votes
2answers
105 views

Asymptotics of a divergent integral

I'm trying to calculate the asymptotics of the following integral: For $\alpha \in (0,1/2)$, $$I(\epsilon) = \int^1_\epsilon s^{\alpha -3/2} \exp \left\{ -\frac{s^{2\alpha -1}}{2} \right\} \exp ...
7
votes
1answer
387 views

How to find asymptotics?

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} ...
3
votes
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 $x \to \infty?$
1
vote
1answer
80 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 ...
2
votes
2answers
52 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 ...
7
votes
3answers
279 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 ...
1
vote
1answer
100 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 ...
10
votes
1answer
304 views

Asymptotic Expansion of an Oscillating Integral

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 ...
8
votes
2answers
218 views

Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]

I am reading about the Riemann hypothesis, and the article mentioned the Li function: $$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$ They said that this function can be approximated: ...
11
votes
1answer
182 views

Calculate Asymptotics of Integral?

Let $f$ be a continuous function on $[0,1]$. How do I calculate the asymptotics, as $n\rightarrow\infty$, of $\displaystyle \int_{[0,1]^n}f\left(\frac{x_1+...+x_n}{n}\right)\text d x_1...\text d ...