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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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1answer
123 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\...
0
votes
1answer
66 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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votes
0answers
30 views

survival probabilty in economics [closed]

I am trying to integrate by parts by using an indicator function. However, I am not really sure if it is a correct way to change the bounds of integral with indicator functions. I am trying to deal ...
-1
votes
1answer
66 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 $r\in[0,1]$...
-1
votes
1answer
74 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 $\...
0
votes
0answers
37 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
votes
0answers
380 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 ...
0
votes
0answers
103 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 ...
5
votes
0answers
159 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
votes
2answers
192 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,...
0
votes
0answers
30 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)=...
0
votes
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) ...
0
votes
0answers
36 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 ...
2
votes
0answers
29 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 ...
1
vote
1answer
167 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
votes
1answer
284 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
161 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
votes
2answers
171 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
votes
0answers
65 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
vote
1answer
76 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,...
2
votes
1answer
193 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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votes
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
67 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
vote
1answer
108 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
244 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
125 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
204 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 ...
0
votes
0answers
67 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^...
2
votes
1answer
124 views

Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule)

The Leibniz integral rule, in its multivariate form, deals with differentiation of the following sort: $$ \frac{\partial}{\partial t} \int_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{...
3
votes
1answer
189 views

Complex integral

We would like to compute (or bound) the following complex integral: $$\int\limits_0^{\infty}\left|\int\limits_{-\infty}^{\infty}\frac{e^{its}}{e^s-\lambda}\,ds\right|\,dt$$ where $\lambda \notin S_{\...
0
votes
0answers
40 views

A nested integral sequence

Let $a_i$, $b_i$, $i=1,\dots n$, be positive numbers and suppose $a_i>b_i$ for all $i=1,\dots n$. Consider the following nested integral $$\tag{$\star$}\label{int} \int_0^x \frac{\cos(x_1)}{a_1-b_1\...
2
votes
2answers
162 views

An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$

For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
17
votes
2answers
836 views

An interesting triple integral

What is the value of this triple integral $$\int\limits_0^{2\pi}\int\limits_0^{2\pi}\int\limits_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz?$$ It has to do with some Schwarz lemma.
3
votes
2answers
246 views

Evaluating the integral $\int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt$

I am trying to evaluate the integral $$ I_k(x)=\int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt $$ with $x$ tending to infinity. In fact, I wish to have an estimate $$ \sum_{k=0}^\infty \frac{1}{\log^k x} ...
9
votes
1answer
416 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}^{...
3
votes
2answers
108 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}...
3
votes
0answers
66 views

Is there a ''simple'' formula for the inverse of the Drinfeld associator?

The Drinfeld associator $\Phi(x_0, x_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x_0, x_1\rangle\rangle$ of formal ...
8
votes
2answers
225 views

Bounding an elliptic-type integral

Let $K>L>0$. I would like to find a good upper bound for the integral $$\int_0^L \sqrt{x \left(1 + \frac{1}{K-x}\right)} \,dx.$$ An explicit expression for the antiderivative would have to ...
0
votes
0answers
171 views

Does $\int_0^\infty f(x+\theta)g(x) \, dx=0\, \forall \theta \in \mathbb{R}$ imply $f=0$ almost everywhere, if $g$ is smooth and strictly positive?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an integrable function, and $g:(0,\infty)\rightarrow(0,\infty)$ a smooth, strictly positive function. If $$\int_0^\infty f(x+\theta)g(x)\,dx=0\qquad\forall\...
0
votes
0answers
85 views

On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let $$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. What are the reasonable asymptotic estimates for $I(T)...
0
votes
0answers
72 views

Gaussian integral over logarithm of shifted error function

Suppose we have the following integral: $$ \int^{\infty}_{-\infty} \frac{dz}{\sqrt{2\pi}}e^{\frac{-z^2}{2} } \log\left( \text{erf} \, a (z-b) +1 \right), \ \ \ \ a,b \in \mathbb{R} $$ Does a closed-...
1
vote
0answers
61 views

If $f$ takes values in $L(H,L(H,\Bbb R))$ and $μ$ is a $H\hat ⊗_πH$-valued measure, how are $\int f\:dμ$ and $\int f⊗_π\text{id}_Hdμ$ related?

Let $H$ be a separable $\mathbb R$-Hilbert space $H\:\hat\otimes_\pi\:H$ denote the projective tensor product of $H$ and $H$ $(\Omega,\mathcal A)$ be a measurable space $\mu$ be a $H\:\hat\otimes_\pi\...
0
votes
0answers
49 views

Fubini theorem two stochastic integrals

The stochastic Fubini theorem is usually understood to be a Fubini theorem for a stochastic and Lebesgue integral. Are there also Fubini theorem for two stochastic integrals? I could imagine ...
1
vote
0answers
92 views

Asymptotics of an integral by two methods

This was asked in MSE, here, but the answer was not satisfactory. I want to compute the asymptotic behavior of the integral $$ f(K,a)=\int_0^1 (1-x)^Ke^{iKa\frac{x}{1-x}}x^2dx$$ when $K$ is large ...
0
votes
0answers
100 views

$\det(I-K(z)+\varepsilon(z,x)) $ versus $\det(I-K(z))$

First let me ask the general question that might interest others dealing with determinantal formulas. We are trying to compare the following two quantities $$C_{\varepsilon} := \oint \det(I-K(z)+\...
0
votes
1answer
53 views

Doubt filling a step in a derivation

I am reading the following book and there is a step in a derivation I don't quite follow. In equation 27 on page 409 it is claimed $$ \int_0^{\infty}\,\epsilon^{\alpha}(1+\epsilon/x)^{-1}e^{-\epsilon}...
1
vote
1answer
126 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^{\...
6
votes
0answers
99 views

A reference for an integrability property?

In a recent paper of mine (Compensated integrability), I established a functional inequality which has nice consequences. For instance, it contains the isoperimetric inequality, and it gives a new ...
1
vote
0answers
121 views

Changing the order of integration of double integral: references and theorems

The Fubini's theorem states that if we have $ \int_0^{\infty} \int_0^{\infty} |f(t,x)| dt dx$ well defined (i.e. function is absolutely integrable) then we can interchange order of integration: $$ \...
1
vote
2answers
144 views

Explicitly representing a random variable in terms of indicator functions

Motivation: I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula. I want to prove the change of variable formula (you ...