2
votes
0answers
77 views

asymptotic expansion of the integral for large tau

How can I proceed to resolve this integral? $$ c_1\int_{-\infty}^{\infty}{\frac{\cos\left(x\tau\right)}{\left(1 + c_{2}\,x\right)^{\alpha}}}\, \,{\rm d}x $$ where $c_1, c_2$ are positive constants, ...
7
votes
1answer
245 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 - ...
4
votes
1answer
127 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 ...
1
vote
0answers
50 views

Integration vs Summation

I am interested in how one might generally evaluate, or estimate $$G(x)=\sum_{n=1}^{\infty}f(n)x^n-\int_{0}^{\infty}f(t)x^tdt$$ as $x\to1^-$, and for a continuous $f$, and such that the integral ...
3
votes
1answer
29 views

Asymptotic analysis of an integral with growing, highly oscillatory integrand

I recently came across the following integral $$\int_0^{\omega} t^{\frac{n-1}{2}}|\cos(t^n)|\,dt.$$ Here $n>1$ is an integer. I was curious as to what its asymptotic form would be. It seems to be ...
5
votes
3answers
123 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 ...
3
votes
2answers
82 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
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}) ...
3
votes
2answers
52 views

Short argument for asymptotic value of parameter integral

I want to find the main term of the asymptotic expansion for $x\to 0^+$ for $$f(x)=\int_0^{\pi/2} \dfrac{\cos t}{t+x}dt.$$ Now, clearly, the problem is at $t=0$ and the cosine is almost $1$ ...
1
vote
1answer
45 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
53 views

Integral asymptotics

Is there some kind of a variation of the Laplace's method or some other formula for the asymptotics of integrals of a type $$\int_a^bf(x)e^{mp(x)}\cos(mq(x)+x/2)dx, \ m\to\infty.$$ Here $f,p,q$ are ...
4
votes
0answers
58 views

Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?

$$\int_0^m \frac{n^x}{\Gamma(x+1)}dx:n,m \in \mathbb{R}$$ I'm dubious as to whether there's a closed form for the above, if there is I'll be very happy. Otherwise: Is there a closed form for ...
1
vote
0answers
41 views

Asymptotics for exponential integrals

Suppose I have a situation where I want to find an asymptotic expansion as $x \to \infty$ for an integral of the form: $$ \int_{a}^{b} f(t) e^{-\phi(t) x} \mathrm{d}t$$ Let us also suppose that ...
4
votes
1answer
133 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
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 ...
2
votes
1answer
101 views

Asymptotic Expansion of a nearly divergent integral

I want to understand the asymptotic behavior of an integral of the form $$ I_f(\epsilon) = \int_0^1 \frac{\log(1/x)}{\sqrt{x}\sqrt{x+\epsilon}} f(x) dx $$ as $\epsilon \to 0^+$ for a generic function ...
1
vote
1answer
84 views

An integral relating to Bernoulli polynomials

Show that $$\int_{0}^{1}B_{2n+1}(x)(\cot({\pi}x)-2\sin(2{\pi}x))dx{\sim}0$$ where $B_{2n+1}(x)$ is the Bernoulli polynomials.
4
votes
1answer
99 views

Bernoulli number type asymptotics

I find an interesting formula but I can not prove it. Show that $$I_n=(-1)^{n+1}\int_0^1 B_{2n+1}(x)\cot(\pi x) \, dx\sim\frac{2(2n+1)!}{(2\pi)^{2n+1}}$$ where $B_n(x)$ is the Bernoulli Polynomials.
10
votes
4answers
385 views

Singular asymptotics of Gaussian integrals with periodic perturbations

At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$, $$ \int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
5
votes
1answer
182 views

Snags when discovering the asymptotic behavior of an integral

I have trouble in discovering the asymptotic behavior (i.e, the asymptotic expansion) of the following integral: $$\newcommand\abs[1]{\left\lvert#1\right\rvert} \int_0^{\pi/2}\frac{dx}{1+(n\pi+x)\sin ...
14
votes
3answers
221 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 ...
2
votes
2answers
205 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.: ...
5
votes
1answer
255 views

expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$

This question relates to this answer I gave to a question about the integral $$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$ I derived an expansion in inverse powers of $p$ and then ...
6
votes
2answers
852 views

Asymptotic expansion of integral involving modified Bessel-function

I would like to obtain the asymptotic expression for $\alpha \to \infty$ of the following integral $$I(\alpha)=\int_0^\infty\!dx\,x (1 - \cos[2\alpha K_0(x)]) = \int_0^\infty\!dx\, 2x \sin^2[\alpha ...
20
votes
3answers
493 views

Asymptotic expression of an oscillatory integral

Consider the integral $$ f(\alpha,\beta)= \int_0^{2\pi}\,dx \sqrt{1- \cos(\alpha x ) \cos(\beta x)}$$ as a function of the two parameters $\alpha,\beta$. I am interested in the asymptotic behavior ...
13
votes
4answers
448 views

Large $n$ asymptotic of $\int_0^\infty \left( 1 + x/n\right)^{n-1} \exp(-x) \, \mathrm{d} x$

While thinking of 71432, I encountered the following integral: $$ \mathcal{I}_n = \int_0^\infty \left( 1 + \frac{x}{n}\right)^{n-1} \mathrm{e}^{-x} \, \mathrm{d} x $$ Eric's answer to the linked ...