All Questions
Tagged with integration real-analysis
352 questions
0
votes
1
answer
126
views
Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$
For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that
$$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?
- 123
2
votes
1
answer
230
views
Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?
Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
1
vote
2
answers
889
views
Simplify Wasserstein distance between Gaussians with binary cost function
Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
- 6,853
2
votes
0
answers
249
views
Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral
It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$
and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$
where $\operatorname{li} (x)$ ...
0
votes
0
answers
185
views
Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral
The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
3
votes
1
answer
494
views
Help in proving, that $\int_{0}^{\infty} \frac{1}{\Gamma(x)} d x=e+\int_{0}^{\infty} \frac{e^{-x}}{\pi^{2}+(\ln x)^{2}} d x$ using real methods only
I hope this does not seem like a too easy question for Overflow. I would like to find an easier method than mine to prove the above statement for the Fransén-Robinson Constant. My first method was to ...
- 169
1
vote
1
answer
246
views
Limits of a family of integrals
Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals.
QUESTION. What is the value of this limit? It seems to exist.
$$\lim_{n\rightarrow\infty}\int_0^1\frac{(\...
- 43.2k
4
votes
1
answer
351
views
Asymptotic behaviour of function using Fox $H$-function representation
In equation (9) of this paper, it is claimed that the limiting behaviour
$$
\int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk
\sim
\frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
- 85
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
2
votes
0
answers
197
views
Orthogonality relation in $L^2$ implying periodicity
Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...
- 405
-1
votes
1
answer
126
views
Is there a name for this family of integral?
This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.
- 101
3
votes
2
answers
235
views
A reduction problem from $\mathbb{R}^2$ to $\mathbb{R}$
Let $f,g \in L^1_\text{loc}(\mathbb{R})$, with $g \geq 0$, and such that for almost every $(x,y) \in \mathbb{R}^2$, at least one of the following equations is true :
\begin{align*}
f(x) + f(y) + g(...
user140442
4
votes
2
answers
718
views
Integrate $1/(x_1x_2\cdots x_n)^k$ for $1\le x_i \le a$, where product of coordinates satisfies $ b\le x_1\cdots x_n\le c$
I need to integrate
$$
\int\limits_{[1,a]^n} \frac {\chi(\{ b \le x_1 \cdots x_n \le c \})} {( x_1 x_2 \cdots x_n)^k} \,dx_1 \cdots dx_n,
$$ where $\chi(E)$ is the characteristic function of a set $E$....
- 91
2
votes
1
answer
83
views
Integral substitution involving the length and angle of two vectors
Let $F\colon\mathbb R^3\to\mathbb R$ be a compactly supported smooth function. I want to compute
$$ \int_{\mathbb R^n}\int_{\mathbb R^n} F(\lVert x\rVert^2,\lVert y\rVert^2,\langle x,y\rangle)~\...
- 623
3
votes
1
answer
363
views
Oscillatory integrals
Consider the integrals
$$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$
I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...
- 4,135
1
vote
1
answer
245
views
Evaluation of a double definite integral with a singularity
How to compute the
$$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy.$$
Is it possible to compute the integral analytically upto some terms. I believe it should involve ...
- 407
9
votes
1
answer
1k
views
Integration by parts formula for the double Riemann-Stieltjes integral
In my research the following integration by parts formula for the double Riemann-Stieltjes integral
$$\int\limits_{[a,b]\times[c,d]}f(x,y)\,dg(x,y)=f(b,d)g(b,d)-f(a,d)g(a,d)-f(b,c)g(b,c)+f(a,c)g(a,c)...
- 3,486
3
votes
0
answers
132
views
A new characterization of Riemann-Integrability
Question :
Given two bounded functions $\,f:[a,b]→\mathbb{R}\,$
and $\;θ:(0,b−a]→[0,1]$.
Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $\,[a,b]$.
Let $\,Δx_k=x_k−x_{k−1}\,$ and $\,\...
- 31
1
vote
2
answers
818
views
Integral formula involving Legendre polynomial
I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...
- 31
6
votes
3
answers
554
views
Computing the volume of a simplex-like object with constraints
For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...
- 61
0
votes
2
answers
178
views
"Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇"
This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
- 723
1
vote
1
answer
2k
views
Expansion of an integral
I have an integral of the form
$$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite ...
- 31
0
votes
1
answer
116
views
Integrable function [closed]
Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...
- 119
2
votes
1
answer
300
views
Necessary and Sufficient conditions for integrable function [closed]
Suppose that $a, b$ and $c$ are constant.
Is there the necessary and sufficient conditions of $a ,b, c$ for the following integration is integrable? i.e.
$$\int_0^\infty \int_0^\infty \int_0^\infty ...
- 119
0
votes
1
answer
350
views
Uniformly Bounded (updating)
Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~
(1+t+z)^{a_3}}\exp\big\{-\frac{...
- 119
-5
votes
1
answer
184
views
a question of definite integral [closed]
1.$$\int_{0}^{1} \frac{1}{1+e^{-(x+\ln(u/(1-u)))/\tau}}\, du$$
2.$$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{+\infty}\frac{e^{-u^{2}/2}}{1+e^{-(x-u)/\tau}}\,du$$
please help me. I tried to use MATLAB but ...
1
vote
1
answer
222
views
Multidimensional improper Riemann integrals with oscillatory kernels: Existence
I have asked this question three weeks ago here
https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930
but received no relevant answers.
Let $n\geq 2$ ...
- 143
1
vote
0
answers
145
views
How to show that this function is continuous (Geometric Measure Theory)
I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by
$$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$
is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
- 19
8
votes
4
answers
1k
views
For what sets does the Lebesgue Differentiation Theorem hold in one dimension?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
4
votes
1
answer
1k
views
Does the Lebesgue Differentiation Theorem hold for regular polytopes?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
2
votes
1
answer
187
views
Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder
I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions
$$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$
$$\int_{\mathbb{R}^+} f \in \mathbb{R}^+$$
...
- 21
5
votes
2
answers
647
views
Dominated convergence 2.1?
After this question : Dominated convergence 2.0?
I want to know, what about the case when $h\in L^1([0,1])$.
The completed question :
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging ...
- 4,074
7
votes
1
answer
552
views
Dominated convergence 2.0?
During my research, I came across the following question.
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that:
$\forall n\in\mathbb N, f_n''<h$, ...
- 4,074
-1
votes
1
answer
149
views
How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]
It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$
Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
- 305
1
vote
0
answers
117
views
Estimation of the integral $\int_a^b e^{2\pi i f(x)} dx $
Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that
$$
\min_{x\in [a,b]} |f'(x)|>\lambda
$$
It is ...
- 675
8
votes
2
answers
330
views
q-Means and the mode of a distribution
Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that
\begin{equation}
\int_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and ...
- 640
10
votes
1
answer
328
views
Asymptotic behavior of an integral depending on an integer
A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where
$$
f(n) := \...
- 1,991
3
votes
1
answer
2k
views
How "compact" are sets of finite measure?
Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover.
Now consider the following situation:
Everything I say in the following is with respect to the ...
user121558
0
votes
0
answers
126
views
$p$-volume of $n$-dimensional hyper-ellipsoids
I read that the unit hypersphere has maximum volume for dimension five and would like to generalize this result. (If you think that integrating over an $n$-dimensional $p$-hyper-ellipsoid area ($x_1^...
- 4,829
1
vote
1
answer
399
views
I want to disprove an equality involving a double integral
I want to show that the following equality does not hold:
\begin{equation}\label{at3}
\frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
- 19
9
votes
1
answer
458
views
Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...
- 43.2k
3
votes
2
answers
179
views
Is $\int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt$ bounded by a constant for large $y$?
For $p\in (0,1)$ and $\epsilon>0$ a small enough constant, consider the function $f:\mathbb{N}\to\mathbb{R}$ given by
$$f(y)=\int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}...
- 43
0
votes
1
answer
139
views
Change of variables for double integral [closed]
Thank you for your time.
My basic question is whether the following change of variables allowed
$$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$
I fail to ...
- 11
1
vote
1
answer
642
views
Interchange of integration order (of a not absolutely convergent integral with sinus)
Can we interchange the integral order of this integral to start integration on $x$ ? (Taking $g$ and $f$ two functions of rapid decrease which are $o(x^2)$ near zero)
$$A=\int_{0}^\infty \int_0^{\...
- 1,199
3
votes
1
answer
237
views
Asymptotics of the following integral
I am concerned with the asymptotic behavior of this integral
$$
\int_2^{\infty}dx\,\frac{\sin(ax)}{ax}\frac{1}{\log x}\bigg(1+\frac{\log x}{\log(a e^{-5/6})}\bigg)^{-\log(a e^{-5/6})}
$$
I am ...
- 41
5
votes
1
answer
881
views
Two integral representations for $\zeta(3)$ from Zurab's integral and standard formulas for the gamma function
This morning I wrote with the help of a CAS, and integral representation for the Apéry's constant $\zeta(3)$ and some standard formulas two formulas involving this constant. I would like to know if ...
user75829
2
votes
1
answer
315
views
Can it be proved that $f$ is integrable?
Let $x$ be a differentiable function on $\mathbb{R}$. I want to prove that for any time $t \geq t_0$
\begin{equation}
\frac{1}{2} D^{\alpha} x^2(t) \leq x(t) D^{\alpha} x(t), \ \ \forall \alpha \...
- 31
2
votes
0
answers
114
views
Is there an explicit version of Morse Lemma used in stationary phase method?
In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
- 3,625
1
vote
1
answer
250
views
Question about the stationary phase method and the smooth function used
A statement of the stationary phase method I know is the following.
Suppose $\phi(x_0) = \phi'(x_0) = 0$ and $\phi''(x_0) \not = 0$. If $\psi$ is a smooth function supported in a sufficiently small ...
- 3,625
15
votes
3
answers
2k
views
Can the Riemann integral be defined through a closure/completion process?
Let us consider real-valued functions on the bounded interval $[0,1]$. A "step function" means an element of the vector space spanned by indicator functions of (points and) intervals in $[0,1]$ (the ...
- 32.5k