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Questions tagged [integration]

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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33 views

How to fix multi-valued function on contour?

I am sorry to ask such an embarrassingly simple question here. My question is about contour integral of the multivalude function. I want to calculate the Fourier transformation of a muti-valued ...
0
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0answers
31 views

Integrability green function

This is maybe a trivial question but i need some clarification to make it clearer in my mind. Consider the fundamental solution of the equation $ \partial_{t} u - \partial^{2}_{xx}u=0$ given by the ...
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1answer
298 views

The derivatives of the integrals with Leibniz Rule? [on hold]

Can we obtain the following result for ... $f(x)={x-\lfloor x \rfloor}$ ... ? Here ${\lfloor x \rfloor}$ is floor function with $a\in \mathbb{R}$ and $u \in \mathbb{R}$ . Thank you for your ...
1
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1answer
76 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 $n\geq 2$ ...
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0answers
42 views

Existence of a function satisfying some integral conditions

I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...
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0answers
25 views

exponential integral of translation invariant function

Consider function $f: (\mathbb{R}^{d})^{n} \rightarrow \mathbb{R}$ with spatial invariance property of the form : $f(x_1,x_2,...,x_n) = f(x_1 + \zeta, x_2 + \zeta,..., x_n + \zeta)$ for $\zeta \in \...
0
votes
1answer
28 views

Probability of a quantity from Ginibre ensemble

I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum_{j=1}^N\lambda_j^2$. So we have $$P(s)=C_{N,K}\int.....
2
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0answers
44 views

Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?

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\...
2
votes
1answer
75 views

Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type $$\int_C f(z) M(\lambda g(z)) dz$$ for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
4
votes
1answer
105 views

Integration of Maurer-Cartan form

Let $G$ be a Lie group with Lie algebra $g$. As it is well known the Maurer-Cartan form $ω:TG\rightarrow g$ transports any vector $X\in T_{x}G$ to the start $l_{x^{-1}*}(X)\in g$, $l_{x^{-1}}$ ...
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3answers
397 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\...
2
votes
1answer
137 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\...
0
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0answers
41 views

What can we say about the Bargmann transform of bounded function?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ Now we define $$ H(t)= H(...
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0answers
114 views

Infintely iterated and functional integration in constructive math

Looking for references on constructive derivations of (elements of) functional integration -- in particular, those used in the classical construction of the Wiener measure. It seems such ...
6
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1answer
165 views

Asymptotic Expansion of Bessel Function Integral

I have an integral: $$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$ and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...
2
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1answer
152 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 \...
3
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0answers
102 views

Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree). I wish to prove or find a counterexample to the following claim: If ...
3
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0answers
56 views

Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
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0answers
151 views

Integral with product of two infinite sums

I am looking for references and results on integrals with product of two infinite sums: $$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$ Above integral is ...
6
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3answers
285 views

Is there a version of Fischer-Riesz theorem for Banach space?

$( \Omega,F, P )$: a measurable space equipped with a finite measure $(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra $p$ : a constant bigger than $1$ ...
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0answers
110 views

How to evaluate these integrals? [closed]

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 ...
4
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2answers
175 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
90 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 ...
4
votes
2answers
142 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
85 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 ...
5
votes
0answers
87 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
61 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 $\...
12
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0answers
413 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|&...
1
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1answer
82 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
228 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
122 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
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1answer
123 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 ...
1
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0answers
87 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
79 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 ...
0
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0answers
76 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
votes
1answer
97 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 ...
1
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0answers
61 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 $\cos(qv^2) \, dv$. ...
9
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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^...
5
votes
2answers
453 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
1answer
383 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
votes
2answers
290 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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votes
1answer
133 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
1answer
77 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
1answer
74 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
90 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
73 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
428 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
0answers
108 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
0answers
177 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. ...