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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4
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0answers
83 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...
2
votes
1answer
508 views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
2
votes
2answers
295 views

Integrating a barycentric monomial over a simplex

Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...
1
vote
0answers
369 views

What does the Riemann–Stieltjes integral measure? [closed]

The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum: $$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$ where ...
11
votes
0answers
248 views

Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
5
votes
1answer
143 views

Asymptotics of Fresnel integrals

It is known that \begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*} is a bounded ...
2
votes
1answer
148 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
6
votes
0answers
98 views

Integral identity related with cubic analogue of arithmetic-geometric mean

This is re-post from MSE as I did not get an answer there. Let $a,b$ be positive real numbers and we define two sequences $\{a_{n}\},\{b_{n}\}$ as follows: ...
1
vote
0answers
104 views

What is $\int (1-e^{-x})^n dx$? [closed]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
7
votes
2answers
438 views

Interesting triple integral

Some time ago I stumbled on an alleged identity $$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= ...
2
votes
1answer
94 views

Asymptotic expansion of a Laplace-type integral with a “manifold of maxima”

Moved this here from MathSE because I had no luck there and suspect the question may be harder than I first thought. Consider the integral $$ I(\alpha)=\int_0^1 dx_1 \int_0^1 ...
2
votes
1answer
204 views

A conjecture regarding the integral of the square of an entire function

Can some help me prove or disprove the following assertion which I encountered in research? Thanks! Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some ...
4
votes
1answer
448 views

Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$ Where s is a complex parameter, and n is a positive integer. The integral ...
4
votes
3answers
376 views

Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...
12
votes
1answer
443 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $$\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $$ has a bunch of integral representations, e.g. ...
4
votes
1answer
240 views

Motivic integration in positive characteristic: how much is known?

It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
2
votes
0answers
59 views

Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved? $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { ...
18
votes
1answer
666 views

Interesting integral

Numerical evidence shows the validity of the following identity $$\int\limits_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$ if $0< ...
2
votes
0answers
217 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
4
votes
2answers
555 views

Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
3
votes
1answer
151 views

Is it possible to get an equation with two exponentials and a bessel function in closed form?

Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...
3
votes
0answers
195 views

A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$: ...
3
votes
2answers
304 views

Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the ...
0
votes
0answers
81 views

Integral over conditioning variable of a Gaussian

The marginal of a multivariate Gaussian can be computed in closed form, i.e., $p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)\ dy$ is simple. But what I need is $L(x) = \int_y \mathcal{N}((x\mid y); ...
1
vote
0answers
336 views

convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral ($z \geq 0$)... \begin{equation} f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = ...
0
votes
0answers
138 views

Asymptotic Expansion of Double integral

Crosspost from math.stackexchange. Have a look at the great answers there, even though they do not quite answer the question completely. Define $$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} ...
2
votes
0answers
228 views

A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
1
vote
1answer
53 views

Expectation of logarithmic of a Laplace random varible

Say $Y$ is a random variable with Laplace distribution with zero mean and variance parameter $b$. I am trying to compute the expectation of $\ln(Y+\alpha)$ ($\alpha>0$), that is: ...
5
votes
0answers
300 views

Reference request : Grothendieck's topological space valued integral

As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
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vote
0answers
90 views

Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
1
vote
0answers
152 views

Reference : Special case of Banach-valued function integration by parts

Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...
2
votes
0answers
215 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
0
votes
0answers
89 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ ...
1
vote
0answers
106 views

Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds : $$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...
0
votes
1answer
93 views

Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?

Suppose I have a 1D advection equation in conservation (divergence) form $\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$ where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
0
votes
0answers
135 views

Integrate Faddeeva function

I came across this integration in my studies. $\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$ It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
0
votes
0answers
439 views

How to solve definite integral involving exponential function

I am trying to get a closed form for the following definite integral: $$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 ...
6
votes
1answer
171 views

Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which $$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$ and $$x_1+\ldots+x_k\leq 1.$$ I am ...
3
votes
0answers
170 views

An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n ...
2
votes
0answers
148 views

Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...
5
votes
2answers
197 views

Integrals involving trigonometric functions and polynomials

Can one describe all the real polynomials $P(x)$ such that the following integrals converge: $$ \int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ? $$ Among special cases are such ...
0
votes
0answers
165 views

Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...
3
votes
0answers
108 views

motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...
0
votes
0answers
164 views

How to evaluate the following integral related to exponential distribution

I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that $$ ...
5
votes
1answer
163 views

The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the ...
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0answers
115 views

Reference: Bochner Integral`

What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
2
votes
0answers
104 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
1
vote
0answers
97 views

How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...
6
votes
0answers
107 views

How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...
7
votes
3answers
527 views

Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath

I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+): $$ \int_{-1}^1\textrm{d}t ...