# 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, ...

981
questions

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vote

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61 views

### Integral average near a point of dispersion

Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...

**0**

votes

**1**answer

195 views

### What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?

Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...

**6**

votes

**1**answer

88 views

### Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that:
If $\lambda\gg 1$...

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votes

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85 views

### Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...

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votes

**0**answers

51 views

### Standard definition: vector-valued essential support

Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used?
$$
\...

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80 views

### Closed form for a double integral over the first quadrant of the $L^p$ disk

Is there, by any chance, a closed form for the following integral
$$
I_p=\iint_{Q_p}(x+y)\log(x+y)dxdy,
$$
where $Q_p=\{(x,y)\in\mathbb{R}^2, x>0,y>0,x^p+y^p\leq1\}$, $0<p\leq\infty$ ?
...

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44 views

### Weak formulation of PDE with weighted inner product

In Boyd's book on spectral methods (available here: https://depts.washington.edu/ph506/Boyd.pdf), I stumbled in section 3.5 "Weak & Strong Forms of Differential Equations: the Use-fulness of ...

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vote

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179 views

### Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...

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votes

**0**answers

85 views

### A conjecture on integrals of infinite products

The problem I would like to discuss in this post is about a conjecture on the following integrals,
\begin{align}
\int_0^\infty \prod_{n=1}^\infty \cos(x/n)\,dx \stackrel{?}= \pi/4 \tag{1}\\
\int_0^\...

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vote

**2**answers

157 views

### Connection between Volkenborn integral and Haar measure on $\mathbb{Q}_p$

This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic.
Since $\mathbb{Q}_p$ is a locally compact ...

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votes

**1**answer

102 views

### Inequality involving Gaussian integral [closed]

I'm looking to prove the following inequality:
$$
\left| \int_0^1 e^{-x^2} \sin(x) \, dx \right| \leq \frac{1}{2} \left(1- \frac{1}{e}\right)
$$
So far I have no idea on how to prove it. Anybody?

**1**

vote

**0**answers

79 views

### L2 norm of the diagonal entries of a random rotation of a fixed matrix?

Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...

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votes

**2**answers

543 views

### Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms*

I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's Table of Integral Transforms vol. i-ii, specifically proofs of the ...

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**0**answers

63 views

### Coincidence Topologies for $L^p$ spaces

If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...

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votes

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541 views

### On properties on a certain functional

Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...

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33 views

### Sufficient and necessary condition for the continuity of an improper integral

Let $f(\cdot) \in \mathscr{C}\left( \mathcal{D}; \mathbb{R} \right)$ where $\mathcal{D} \subseteq \mathbb{R}$ is open with $0 \in \mathcal{D}$ and
$$ f(0) = 0, \quad \forall x \in \mathcal{D}\...

**0**

votes

**1**answer

180 views

### Generalised limits via derivatives of integrals?

Assuming that $f$ is a continuous function, we have that
$$f(x) = \frac{d}{dx}\int f(t)\,dt.$$
Assuming instead that $f$ has a removable singularity at $x=a$, and is otherwise continuous, we have ...

**2**

votes

**2**answers

189 views

### Asymptotic decay rate of an oscillatory integral

Consider the following oscillatory integral
$$
I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $...

**2**

votes

**0**answers

74 views

### The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse

Conjecture.
Let
$$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$
Then for any ...

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90 views

### Is this integral finite and how does it decay to zero?

I would like to know if the following is convergent/finite (it represents a bound from a truncated Legendre series approximation)
\begin{equation}
\varepsilon_n \leq \int_{-1}^1 \left(\int_{n\gg 1}^\...

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vote

**0**answers

24 views

### Convolve a 4D Gaussian function along a plane?

There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector.
Now I want to blur (convolve) it along with $u$ by another 2D ...

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vote

**0**answers

26 views

### Spectrum of large random asymmetric matrices with correlation

Background:
In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices $\mathbf{J}$ by studying the following integral:
\begin{equation}
I=\left[\...

**2**

votes

**1**answer

152 views

### Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...

**3**

votes

**0**answers

128 views

### Accuracy of Richardson's error estimate in the presence of rounding errors

Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter $h$. Common ...

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votes

**1**answer

116 views

### Evaluation of $\int \prod_{j=1}^u \frac{x+j}{j-x}~dx$

Let $I_{n,k} = \frac{(n+k)!}{k!(k-1)!(n-k)!}$. This is a sort of generalization of the Apéry's numbers, with $I_{n,n} =$ the $n$-th Apéry number. I am studying integrals of the form:
$$f_u(x)=\int \...

**6**

votes

**1**answer

264 views

### Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$

Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to ...

**0**

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**1**answer

121 views

### Orlicz-Sobolev Spaces

let $A$ an N-function and suppose that
$$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty $$
we denote by $\widehat{A}$ an N-function equal to A near infinity and $\widehat{A}$ ...

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**1**answer

82 views

### Sign of expectation value

Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...

**3**

votes

**1**answer

131 views

### $L_p(I,Y)^\perp=L_q(I,Y^\perp)$?

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. For $1<p<\infty$ consider the $p$-th power Bochner Integrable functions which takes values in $X$ and defined on the unit interval $...

**34**

votes

**1**answer

1k views

### Existence and uniqueness of Haar measure on compacta; a cohomological approach

I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...

**2**

votes

**0**answers

184 views

### The collection of mean value abscissas in the Mean value theorem

The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...

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votes

**1**answer

204 views

### Fubini's theorem on arbitrary foliations

In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...

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125 views

### Functions that are Khinchin integrable but not Henstock-Kurzweil integrable

I posed this question on Mathematics SE recently, though by the total lack of attention it has gotten, I do not anticipate an answer and bring it here.
What are some Khinchin integrable $f$ which ...

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**0**answers

85 views

### Conceptual meaning of a non-linear relation connecting $6$ Mordell integrals?

Define Mordell integral by
$$
\phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1}
$$
There are a lot of linear relations connecting integrals ...

**0**

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48 views

### Expression of divergent integrals via simplier divergent integrals (or series)

Yesterday I just came to a formula that may allow to express divergent improper integrals bounded (on any finite interval) functions in terms of simple improper integrals or series.
$$\int_0^\infty f(...

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120 views

### Why is the divergence theorem used in the Eells-Sampson paper slightly different from that in a textbook?

I am reading Harmonic Mappings of Riemannian Manifolds by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for ...

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**0**answers

140 views

### The Poincaré Lemma

Let me consider an $L^1(\mathbb R^N)$ function $f$ such that $$
\int_{\mathbb R^N} f(x) dx =0.
$$
Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector ...

**1**

vote

**1**answer

211 views

### How can we do a Gaussian integral over matrix elements?

I am integrating the following Gaussian over all possible matrix elements $J_{ij}$:
$$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{...

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votes

**1**answer

87 views

### Integration of a particular rational expression [closed]

I am trying to solve the following integration, where $a,b,c,d,e$ and $f$ are constants:
$$I=\int\frac{x^4+ax^3+bx^2+cx+d}{x^3(x^3+ex+f)}dx$$
I tried to solve the integral using the following two ...

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votes

**3**answers

207 views

### Is there a good approximation for this Gaussian-like integration?

Is there an analytic solution or approximation for the following Gaussian-like integration? $\frac{1}{\eta^{2n}} \frac{1}{\sqrt{2 \pi}} \int_{-\eta}^{+\eta} e^{-x^2/2} x^{2n} dx$? The numerical plot ...

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votes

**0**answers

114 views

### Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...

**1**

vote

**0**answers

169 views

### Adaptive Simpsons Quadrature Algorithm for Double Integrals? [closed]

I'm currently using Numerical Analysis 10th edition by Richard L Burden as a reference for approximate Integration techniques. In there it describes the Adaptive Simpsons Quadrature rule that inputs ...

**0**

votes

**1**answer

200 views

### Integral $ g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $ [closed]

I am having trouble calculating this integral:
$$ g(a) = \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $$
I tried calculating $g'(a)$ but then I get stuck.

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votes

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71 views

### Can we solve this integral equation?

Let $(E,\mathcal E,\lambda)$ be a measure space, $p,q_i$ be positive probability densities on $(E,\mathcal E,\lambda)$ for $i=1,2$, $\mu:=p\lambda$, $\sigma_{ij}:E^2\to[0,\infty)$ be $\mathcal E^{\...

**1**

vote

**0**answers

79 views

### Is harmonic mean of linear functions a Bernstein function?

According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function:
$f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$
is a Bernstein function, ...

**1**

vote

**0**answers

77 views

### Expressing 1-e^{-z} as a Fourier integral

According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...

**0**

votes

**0**answers

38 views

### moment generation function for matrix Gaussian distribution

What is the moment generation function for the following function
$$
E(e^{\mathbf{X}^T\mathbf{y}\mathbf{W}})=\int e^{\mathbf{X}^T\mathbf{y}\mathbf{W}}\frac{\exp\big(-\frac{1}{2}\mathrm{tr}\big[\mathbf{...

**1**

vote

**1**answer

132 views

### Integration of a particular quartic form

I would like to solve the following integral:
\begin{equation}
\int \prod_i d x_i e^{a x_i^2 + b x_i^4 + c x_i^2 x^2_{i+1}}
\end{equation}
This integral can be for sure lead back to a common ...

**0**

votes

**1**answer

130 views

### Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$

I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...

**7**

votes

**1**answer

646 views

### On the closed-form of $\int_0^1\int_0^1\int_0^1\frac{dxdydz}{1-\frac{z}{3}(x+\sqrt{xy}+y)}$

I would like to know if it is possible to calculate in closed-form, or well what work can be done about it, the definite integral $$\int_0^1\int_0^1\int_0^1\frac{3dxdydz}{3-z(x+\sqrt{xy}+y)},\tag{1}$$
...