# Tagged Questions

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
...

**3**

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**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**0**

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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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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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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 ...

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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**

votes

**1**answer

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})\...

**1**

vote

**1**answer

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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**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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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**1**answer

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

**1**answer

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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**0**answers

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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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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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
$$ \int_{x_0}^{x_1} dx \cos(ax) \frac{\sum_i^N w_i(x) e^{-b_i^2 x^2 + c_ix}}{\sum_i^N w_i(x)},$$...

**2**

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**1**answer

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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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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**2**answers

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^...

**5**

votes

**2**answers

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 ...

**7**

votes

**1**answer

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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**2**answers

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 ...

**-1**

votes

**1**answer

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\...

**1**

vote

**1**answer

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 ...

**-1**

votes

**1**answer

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 $r\in[0,1]$...

**-1**

votes

**1**answer

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 $\...

**0**

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**0**answers

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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**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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,...

**0**

votes

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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)=...

**0**

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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

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

vote

**1**answer

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,...

**2**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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

**0**answers

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**

vote

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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
...