Tagged Questions

7
votes
3answers
176 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
31 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 ...
8
votes
2answers
100 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: ...
10
votes
1answer
139 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 ...
0
votes
0answers
42 views

Asymptotically expand Laplace transform

Assume $\max_{a\leq t\leq b}{\phi (t)} =\phi (a)$, $\phi '(a)\neq 0,f(a)\neq 0$ and $f(t)$ has a Taylor expansion about $t=a$. Use integration by parts to show that $I(x)=\int_a^b{f(t)e^{x\phi ...
12
votes
3answers
165 views

Sufficient bound to conclude limit has certain value. $\lim {\left( {\int_0^1 {{{dx} \over {1 + {x^n}}}} } \right)^n}=\frac 1 2 $

I am trying to show that $$\lim {\left( {\int\limits_0^1 {{{dx} \over {1 + {x^n}}}} } \right)^n}=\frac 1 2 $$ Now, this can be done as follows. Using $x\mapsto x^{-1}$ we get that $$\int\limits_0^1 ...
0
votes
1answer
70 views

Asymptotic of a particular integral

This is in relation to my question here. I am reading from this paper and specifically the doubt is from a statement on page 177. Suppose $\alpha\in(0,2)$ and $t_i=ih$ for some fixed $h>0$ and ...
0
votes
1answer
87 views

Meaning of algebraic decay

I am reading the paper here and I am running into a few roadblocks. One of them was resolved here and now I am stuck at another. (Pg 177) Suppose $\alpha\in(0,2)$ and $t_i=ih$ for some fixed $h>0$ ...
3
votes
1answer
82 views

Laplace method help

$$\int_{0}^{\infty} \frac{e^{-x \cosh t}}{\sqrt{(\sinh t)}}dt$$ I'm trying to use Laplace's method to find the leading asymptotic behavior as $x$ goes to positive infinity, but I'm having some ...
5
votes
0answers
126 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega ...
10
votes
2answers
186 views

Laplace's method

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: ...
2
votes
2answers
89 views

$f(x)=\int_{0}^{1}e^{ixz^2}dz$ as $x\rightarrow +\infty$.

Find the asymptotic behaviour as $f(x)=\int_{0}^{1}e^{ixz^2}dz$ as $x\rightarrow +\infty$. Could anyone show me how to do this with either the method of stationary phase or integration by parts? ...
3
votes
1answer
100 views

Asymptotics of exponential integral

Hello I wonder if there is any asymptotics known for such integral: $$ I(x) = \int_2^x \frac{e^t}{t} dt \qquad\text{when $ x\to+\infty $}. $$ Thank you very much.
1
vote
1answer
151 views

Asymptotic expansion of integral

Consider the following integral: $$ \int_{0}^{\infty} e^{-xt} \ln(1+\sqrt{t})dt $$ Calculate its asymptotic expansion to ALL orders as $x\rightarrow\infty$. It seems the natural thing to do is ...
4
votes
1answer
107 views

Laplace integral and leading order behavior

Consider the integral: $$ \int_0^{\pi/2}\sqrt{\sin t}e^{-x\sin^4 t} \, dt $$ I'm trying to use Laplace's method to find its leading asymptotic behavior as $x\rightarrow\infty$, but I'm running into ...
0
votes
1answer
58 views

Asymptotic evaluation of integral

I am studying how to evaluate the integral $$\int_{0}^{\pi/4}{d\theta \over \epsilon^2+\sin^2\theta}$$ as $\epsilon \rightarrow 0$ with asymptotic methods. The book: perturbation methods by Hinch ...
15
votes
2answers
265 views

Asymptotic analysis of the integral $\int_0^1 \exp\{n (t+\log t) + \sqrt{n} wt\}\,dt$

The integral I'm trying to study is $$ F(n) = \int_0^1 \exp\left\{n(t+\log t)+\sqrt{n}wt\right\}\,dt, \tag{1} $$ where $w$ is a fixed complex number with $\Re(w) < 0$ and $\Im(w) > 0$. As ...
1
vote
1answer
101 views

convergence of an oscillatory integral

Let $\alpha$ be real numbers and let $f\colon\mathbb{R}\to \mathbb{C}$ be a function in $L^2 (\mathbb{R})$ (actually smooth and compactly supported, but this doesn't seem to be relevant). I am ...
2
votes
2answers
131 views

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: ...
1
vote
4answers
127 views

Estimate on an oscillatory integral

During my research I ran into the following type of an oscillatory integral, for some values of nonzero reals $a,b$: $f(R):=\int_{0}^{R} e^{2 \pi i (ar^2 + br)} dr$ and I am interested in finding a ...
7
votes
2answers
139 views

Equivalent of $\int_0^{\infty} \frac{\mathrm dx}{(1+x^3)^n},n\rightarrow\infty$

According to my calculations $$ \int_0^\infty \frac{\mathrm dx}{(1+x^3)^n}=\frac{(3n-4)\times(3n-7)\times\cdots\times5\times2}{3^{n+1/2}(n-1)!}2\pi$$ How can an equivalent of $$ \int_0^\infty ...
0
votes
1answer
99 views

asymptotic limit of $\int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt$

Help me please with the following integral. I've asked this question before Asymptotic limit of the integral with polynomial, but it turns out that it was incorrect question. I should get an ...
1
vote
2answers
204 views

Bounds on integral $x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy$

Consider the function $$ I(a,x) = x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy $$ where $x \geq 1$, and $a \geq 0$. I am not really interested in the parameter $x$, so define $$ I(a) = \sup_{x \geq 1} ...
3
votes
1answer
199 views

How does one integrate Landau symbols?

I have some big O()'s in an integral. How can i compute or estimate such an integral?
2
votes
2answers
168 views

Simplifying the generalized function $x^{\lambda}_+$ in the strip $-n - 1 < \mbox{Re}\lambda < -n$

Note: this post is a follow up to an earlier question. The (divergent) integral of $x^{\lambda}_+$ can be analytically continued into the region Re $\lambda > -n - 1$, $\lambda \ne -1, -2 , \ldots ...
2
votes
2answers
114 views

Series expansion of $ \int_{0}^{1} \frac{x^n}{1+x^n} \mathrm dx $

I have proved $$ \int_{0}^{1} \frac{x^n}{1+x^n} \mathrm dx \sim \frac{\ln(2)}{n}$$ How can I get further and find $ a$ such that: $$ \int_{0}^{1} \frac{x^n}{1+x^n} \mathrm ...
6
votes
1answer
169 views

Showing that $\int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx$ is convergent for $\lambda > -2$

Id' appreciate help understanding why the integral $$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx $$ is convergent provided $\lambda > -2$, where $\phi \in \mathcal{D}(\mathbb{R})$. To ...
2
votes
1answer
122 views

Asymptotic expansion of an integral

Here is an exercise from Dieudonné. He suggests to "perform integrations by part". Let $f, g$ be positive $C^\infty$ functions, $F(x)=\int_1^x f(t)dt$ and assume that $\int_1^\infty f(t) dt = ...
4
votes
1answer
118 views

How to estimate this integral

How to estimate the following integral: $$\int_e^x \log{\log{t}} dt$$ so that the error term is within $$O\left(\frac{x}{\log^2{x}}\right)$$. Assume $$x>e$$ Any hint?