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Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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97 views

How to evaluate these integrals? [on hold]

This is a problem of ordinary calculus. Given \begin{align} f(x)&= \left\{ \begin{array}{ccc} k(0),&\quad 0\leq x<b\\ k(x-b),& \quad b\leq x<b+a ...
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0answers
102 views

Integral equality of 1st intrinsic volume of spheroid

Computations suggest that $$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$ The question ...
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0answers
68 views

An exponential integral over sphere

I'm wondering if someone knows if the following kind of integral has a closed-form expression (or maybe an approximation): $$\int_{0}^1 x^{2a} (1-x^2)^b e^{-x^2 c} dx $$ for $a,b,c > 0$. [This has ...
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0answers
27 views

Symmetric double integral expression

Let us consider an expression of the form $Y=\int\int dx dx' f(x)f(x')G(x,x')$ where $G(x,x')$ is a Green's function and $f(x)$ real valued function. Can we obtain any simplification of $Y$ (e.g. to ...
3
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0answers
28 views

Convergence of the intertwining operator as a vector valued integral

Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \...
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0answers
80 views

Gelfand-Pettis integral: what does it mean for a topological vector space to “admit a dual space?”

I am trying to understand more about the Gelfand-Pettis integral. From wikipedia: What does it mean that $V$ "admits a dual space?" When $V$ is a Banach space, $V^{\ast}$ is taken to be the space ...
4
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0answers
80 views

Remainder term in an integral linked to the Riemann zeta function

Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have $\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
3
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1answer
58 views

Echange of Infimum Integral with Pointwise Infimum

Setup Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
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1answer
50 views

Solving integral over the space of two lognormally distributed variables

I am interested in the closed form solution to the following problem: $\int_a^b\int_0^{cx+d}(x+e)f(x)f(y)dydx$, where $f(.)$ is the pdf of the lognormal distribution with mean $0$ and variance $\...
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402 views

Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint. It is not difficult to see that $$\int_{1<|z|&...
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0answers
78 views

Does this sequence have a convergence subsequence?

Let $w_n\in C([0,\tau];L^2(\Omega))$, and $\Omega$ be an open bounded set of $\mathbb{R}^2$. For every $t\in [0,\tau]$, $w_n(t)$ has a convergent subsequence in $L^2(\Omega)$. Does the sequence $$\...
1
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1answer
79 views

Is continuity preserved under norm operations

Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...
6
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1answer
217 views

Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$ is not ...
3
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1answer
112 views

Gauge integral versus path integral

According to this paper, "The gauge integral [a.k.a. Henstock-Kurzweil integral] provides the only formal framework that is close to the original development of the Feynman path integral", and also "...
3
votes
1answer
109 views

Computing relative cohomology class of differential form

When dealing with a top degree differential form $\mu$ in a manifold $M$, a way of "computing" its cohomology class is integrating it through the whole manifold. For instance, if the integral $ \int_M ...
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0answers
85 views

Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
3
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0answers
73 views

Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

I have the following multiple integral derived from a random matrix calculation I wish to evaluate $$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$ where the $\rho$ functions ...
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0answers
73 views

Integral of function with sum of gaussians in denominator

Edit: Added subscript i to b and c in integrand I need to integrate a function with the following form $$ \int_{x_0}^{x_1} dx \cos(ax) \frac{\sum_i^N w_i(x) e^{-b_i^2 x^2 + c_ix}}{\sum_i^N w_i(x)},$$...
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1answer
86 views

Multiple Wiener-Ito integral distribution

Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$ Is it possible to find the distribution of multiple Wiener-Ito ...
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0answers
57 views

Fresnel Integral and his formulas

Below is how Fresnel approximate the eponymously "Fresnel Integral". In his own words: Let $i$ and $i+t$ be the narrow limits between which it is proposed to integrate $\cos(qv^2) \, dv$. ...
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2answers
1k views

Difficult trigonometric integral

This question was also asked here and here. I have faced some difficulties to do the following integral: $$ I=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta~\sin\theta\int_{0}^{\infty}dr~r^2\frac{3x^2y^...
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2answers
340 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 ...
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1answer
371 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: $\forall n\in\mathbb N, f_n''<h$, ...
3
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2answers
284 views

Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove: $$ \int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1. $$ Numerically it seems to hold true. So I have made some attempts to ...
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1answer
128 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\...
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1answer
76 views

Compatibility of the absolute value with the integration

Let $H$ be a separable Hilbert space and $L^{1}(H)$ be the space of trace class operators on $H$. Let $f:\mathbb{R}\to L^{1}(H)$ be a measurable function that is $t\to \operatorname{tr}(f(t))$ is ...
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1answer
71 views

A smooth curve and mean value theorem

Assume that $f$ is smooth function defined in the unit disk $D: x^2+y^2\le 1$, and consider the integral $$I=\int_D f dxdy=\int_0^1r \int_0^{2\pi} f(re^{it})dt.$$ Then it is clear that for $r\in[0,1]$...
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1answer
82 views

A non-trivial upper bound on the integral of Lipschitz functions over a bounded support

Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. Let $f(0)=0$. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\...
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0answers
48 views

Integral of the product of Normal density (PDF) and CDF with limits

Similar to a previous post, I need to integrate the product of a density (PDF) and a CDF, but this time in just the nonnegative domain. My equation is of the form: $$\int_{0}^{\text{∞}}\Phi(-\alpha x)\...
10
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0answers
414 views

multi-dimensional integral of modified Vandermonde determinant

I'm looking for suggestions on how one might try to compute the following $(N-1)$-dimensional integral: $$I_N= \frac{1}{(2\pi)^{N-1}(N-1)!} \int\cdots\int \\ \begin{vmatrix} 1 ...
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0answers
106 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 ...
6
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170 views

Basel problem and inversive geometry

An interesting solution of the Basel problem is given in https://www.tandfonline.com/doi/abs/10.1080/00029890.2018.1452473 (Basel Problem: A Solution Motivated by the Power of a Point, by Kapil R. ...
4
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2answers
197 views

Elliptic-type integral with nested radical

Let: $$Q(A,Y) = Y^4-2 Y^2+2 A Y \sqrt{1-Y^2}-A^2+1$$ I’m curious as to whether it’s possible to find a closed-form solution for: $$I(A)=\int_{Y_1(A)}^{Y_2(A)} \frac{1}{\sqrt{\left(1-Y^2\right) Q(A,...
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0answers
35 views

A two argument sepcial function related to Legendre polynomial and Meixner polynomial

This problem raised when I was trying to evaluate a complicated integral. A polynomial with 2 arguments emerged and I could not recognize it. Let's call it $F_n(k,x)$, what I know is that $F_n(0,x)=...
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0answers
71 views

Can you super-integrate general power function?

Let $ \theta \in \mathbb{R}^{0|1} $. Then the super-integral \begin{equation}\label{superint} I (p) \, : = \int \theta^p d\theta , \end{equation} is well defined for $ p \in \mathbb{N} $, with $ I (0) ...
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0answers
42 views

Interchanging limit and integral and partial sums involving characteristic function

Let \[ B_{N}(t) \equiv \sum_{k \leq t} a(n)\chi_{A_{N}}(n), \] where $A_{N}$ is some subset of $\mathbb{N}$ such that $A_{N} \to \mathbb{N}$ as $N \to \infty$ and $\chi_{A_{N}}(n)$ is the ...
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0answers
36 views

integration by parts for fractional laplacian formula for a larger class of functions

When is this formula valid $\int_{\mathbb R^N} (-\Delta)^sf g =\int_{\mathbb R^N} (-\Delta)^s g f $ with $s\in (0, 1)?$ I am aware of the integration by parts formula holds for $f, g$ in $L^2(\mathbb ...
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1answer
168 views

Example of a smooth function in a manifold whose integration vanishes [closed]

Let $M$ be a complete Riemannian manifold. Now for a fixed $p\in M$, is there any non-constant smooth function $u:M\rightarrow\mathbb{R}$ such that $$\int_{B_r}udV=0\ \forall 0\leq r<\infty,$$ ...
10
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1answer
289 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) := \...
2
votes
1answer
216 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 ...
4
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2answers
201 views

Invariance of an integral over SO(3) under permutation of parameters

The integral $$Z_3(\lambda_1,\lambda_2,\lambda_3)=\frac{1}{2}\int_{-1}^1 I_0\left[\tfrac{1}{2}(\lambda_1-\lambda_2) (1-x)\right] I_0\left[\tfrac{1}{2} (\lambda_1+\lambda_2)(1+x)\right]\,e^{\lambda_3 ...
5
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0answers
66 views

Hilbert series for invariant ring

I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with). To put it in some broader context, let $G$ ...
1
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1answer
78 views

Difficul integral- Solow model [closed]

My friends are preparing project about a Solow model. The asked me to calculate such integral: $ s(1-a)\int e^{(1-a)(b+c)t} \cdot (d-ge^{ft})^{1-a}dt$ where: $b,c,s,d,g>0$ and $a∈(0,1)$. $[a,b,c,d,...
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1answer
200 views

Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as ...
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0answers
55 views

Approximate by by a function $h_{n}\in C^{\infty}(\Omega\times [0,T]\times\mathbb{R})$

Let $\Omega$ be a bounded smooth subset of $\mathbb{R}^n$ and $T>0$ fixed. And let $h$ be a function defined on $\Omega\times [0,T]\times\mathbb{R}$ with values in $\mathbb{R}$. For almost ...
4
votes
0answers
69 views

Computing the volume of intersection between a hyper-rectangle and a ball

$C$ is a region bounded above, below coordinate-wise by $\overline{c},\ \underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex with $d$ that computes the volume of $C\cap B(0,1)$ ...
1
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1answer
131 views

integrate exponential function of quadratic form over unit norm vectors

Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively. I need to solve $$\int_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right) \, dx$$ where $(\cdot )^\...
3
votes
1answer
250 views

Bounding a series of nested integrals

Consider the following matrix function $$ f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0, $$ where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers. ...
4
votes
1answer
127 views

$L^2$-valued integral as parameter integral

Setting Let us regard the Hilbert space $L^2(0,1)$ and the $C_0$-semigroup $(T(t))_{t\geq 0}$ defined by $$ T(t):\left\{ \begin{array}{rml} L^2(0,1) & \to & L^2(0,1), \\ [f]_{\sim} &\...
5
votes
2answers
219 views

An integral involving three Bessel functions

I am looking for a closed form for the following integral $$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$ which can be thought of as a particular case of the more general integral ...