# 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, ...

923
questions

**2**

votes

**1**answer

50 views

### $ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space.
The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by:
$$
\...

**3**

votes

**1**answer

206 views

### Integral convergence implies pointwise

This is a simplified version of a question I am looking at, but embarrassingly I can't do this one anyway.
Let's assume $f(x)$ is a decreasing positive function, $f(0)$ is infinite and, moreover, $f(...

**-3**

votes

**0**answers

28 views

### Prove the following path integral in the complex plane [closed]

enter image description here
The image contains an equation of a path integral.

**-3**

votes

**0**answers

107 views

### Integration modulo integers

$f(x,\theta)=\frac{g(x,\theta)}{h(x,\theta)}$ be a function parametrized by $x\in\mathbb N$ such that its integral with $\theta\in[a,b]$ for fixed $a,b\in\mathbb R$ is always an integer.
The ...

**4**

votes

**0**answers

144 views

+50

### Batalin-Vilkovisky integral is invariant under infinitesimal deformation

This is from page 90 and 93 of Mnev's paper BV formalism and applications.
Let $\mathcal L_{t} \subset \Pi T^{*}M$ be a smooth family of Lagrangians with $t \in [0,1]$ a parameter, s.t.
$\...

**0**

votes

**0**answers

35 views

### Find the volume of the solid obtained by rotating the region [closed]

Let R be the region in the first quadrant bounded by the curves y = f(x) = 2x+ 1
and y = g(x) = 2x^2 − 8x + 9
Find the volume of the solid obtained by rotating the region R about
y-axis using two dy-...

**0**

votes

**1**answer

76 views

### Asymptotic development of Integral of $e^xx^r$

Let $\alpha \in (0,1)$ and $\delta \in (0,1/2)$ be fixed, and consider the following integrals for each integer $j \geq 0$:
$$I_j(u):= \frac{e^u}{u^{j+\alpha}} \int_{-u\delta}^0 e^t t^{j-1+\alpha}\...

**0**

votes

**1**answer

45 views

### Limit of the convolution of derivative of Gaussian heat kernel

I'm looking for the following limit:
$$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...

**-2**

votes

**0**answers

43 views

### Range of i.i.d random variables [closed]

The density funtion of the range of n identically distributed random variables taking values in $(a,b)$ with the common cumulative distribution function $F(x)$ and pdf $f(x)$ is known to be given by $$...

**0**

votes

**0**answers

37 views

### Integration of exponential of square root of quadratic polynomial

I know that
$\int_0^\infty \exp(-\sqrt{x^2 + b^2}) \mathrm{d}x = b K_1(b)$,
where $K_1$ denotes the modified Bessel function of second kind and $\mathrm{Re } b > 0$ (Gradshteyn-Ryzhik 7th ed., 3....

**22**

votes

**1**answer

387 views

### A characterization of constant functions

In How to recognize constant functions. Connections with Sobolev spaces (Russian Math Surveys 57 (2002); MSN), H. Brezis recalls the following fact:
Let $\Omega\subset{\mathbb R}^N$ be connected ...

**0**

votes

**0**answers

63 views

### How to integrate an exponential function of a rational function?

Can anyone help me to calculate the following integral?
\begin{align}
\int\limits_0^t {{{(x - t)}^2}} x\,{e^{ - \left(x + \frac{a}{{bx + 1}}\right)}}\mathrm{d}x
\end{align}
where $a$ and $b$ are ...

**2**

votes

**1**answer

102 views

### An interpolation inequality

I am interested in the following statement. Let $q>p$. Then there are positive numbers $\alpha$ and $\beta$ so that for all $f\in C^1(\mathbb{R}^n)$, one has
$$ \left(\int|\nabla f|^p dx\right)^\...

**-6**

votes

**0**answers

262 views

### Analyzing my definition of Average which uses a variation of the Lebesgue Integral and Measure [closed]

Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. As noted in previous questions, I want to define an average using a new measure and integral since I found certain aspects of the ...

**0**

votes

**0**answers

38 views

### Interpret certain expressions in terms of classical quadratic surfaces

I have two primary constraints, a linear one,
\begin{equation} \label{C1}
C_1=Q_1>0\land Q_2>0\land Q_3>0\land Q_1+3 Q_2+2 Q_3<1,
\end{equation}
and a quadratic one (incorporating $C_1$),
\...

**1**

vote

**0**answers

119 views

### A question about rationality, irrationality or transcendence of definite integral [closed]

Forgive me for the following fundamental question. But I think I require the accuracy of an expert.
Consider an integral of the form:
$$\int_a^b f(x)dx,$$
where $f(x)$ is analytic and real valued for ...

**3**

votes

**0**answers

53 views

### References on integration on non-compact manifolds

I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem.
In particular I have non-compact manifold $M$ and I have an integral (in ...

**-5**

votes

**2**answers

418 views

### Coming up with a equivalent (or close) definition for an average which is easier to compute? [closed]

Continuing from my last question, I understand that my definition is unclear so I have modified it.
Since no one has answered my question on math stack exchange, I decided to ask here.
Definition
...

**2**

votes

**0**answers

31 views

### First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space

The curve $\Gamma$ in $\mathbb{R}^2$ is defined by a continuous and monotonically increasing function $f(x)\in\text{C}[0,1]$, where $f(0)=0$, $f(1)=1$.
Let $(X,Y)$ is jointly and uniformly ...

**-2**

votes

**1**answer

422 views

### Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]

Edit: I got rid of my old definitions. Everything should be clear now
Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...

**6**

votes

**1**answer

108 views

### When is the Radon-Nikodym derivative locally essentially bounded

Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I_K)<\infty$ for every ...

**0**

votes

**0**answers

40 views

### Integral of an expression including a fraction having modified Bessel functions of the first kind on both numerator and denominator

I am looking for an analytic result of the following integral
$$\iint_0^\infty {{\rm{d}}x{\rm{d}}y{x^2}{y^2}\exp \left\{ { - {\alpha _1}{x^2} - {\alpha _2}{y^2}} \right\}\frac{{{\rm{I}}_1^2\left[ {\...

**2**

votes

**0**answers

61 views

### Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology

This is related to these posts and here.
Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...

**0**

votes

**0**answers

50 views

### Integral with 4 Bessel functions and an exponential

I would like to solve the following integral
$$
\int_0^\infty e^{-a k^2} J_{3/2}(b k) J_{3/2}(c k) J_{3/2}(f k) J_{1/2}(r k) k^{-3} dk,
$$
where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...

**4**

votes

**0**answers

73 views

### Is there a known closed form expression for this integral?

I am interested in the following integral:
$$f(x,y) = \int_{\mathbb{S}^d} \max(0,x^Tw)\cdot\max(0,y^Tw) \, dw, \qquad x,y\in\mathbb{S}^d,$$
where $\mathbb{S}^d\subset\mathbb{R}^{d+1}$ is the $d$-...

**3**

votes

**1**answer

98 views

### Euler–Maclaurin formula in $\mathbb{Z}^d$

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...

**0**

votes

**1**answer

75 views

### Integral rising from difference of chi-squared random variables

Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq ...

**2**

votes

**0**answers

104 views

### How do I evaluate the following double integral?

I would like to evaluate the following double integral:
$$
\int_{-1}^1d\zeta\int_{-1}^1 d\bar{\zeta} (\zeta+\bar{\zeta})^{d-2}[(1+\zeta\bar{\zeta})(\zeta-\bar{\zeta})]^J \,\times [(1-\zeta)(1+\bar{\...

**2**

votes

**0**answers

26 views

### An exponential integral over a closed convex polytope

For any $T\geq 2$, let us define the polyhedron $S$ given by
\begin{align*}
S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\}
...

**6**

votes

**2**answers

301 views

### Why is $-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx$ equal to $\phi^2$?

I came across this integral involving the derivative $f'(x)$ of the Fermi function $f(x)=(1+e^x)^{-1}$:
$$I(\phi)=-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx.$$
I'm pretty certain ...

**2**

votes

**1**answer

50 views

### fractional moments of multivariate normal distributions

is there an analytic formula for fractional moments of multivariate normal distribution?
$E(\prod_{i=1}^k x_i^{\nu_i})=?$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and $\...

**0**

votes

**0**answers

99 views

### Finding a square integrable dominating function for function class

problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$
where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...

**0**

votes

**1**answer

78 views

### $ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.
Suppose $\{X_n\}$ is a sequence of random variables satisfying :
$$
\sup_{n}{\mathbb{E}(|X_n|)} <\infty
$$
Suppose that
$$
\dfrac{M_j}{...

**2**

votes

**1**answer

84 views

### Duality form of $L^q$ norm, without assumption that $\int fg$ defined?

The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden.
Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space.
Let ...

**3**

votes

**2**answers

113 views

### Meinardus theorem at use: problems with conditions

I am working on an enumerative problem related to knot theory, and I have found the following generating function
$$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$
I am interested on getting ...

**1**

vote

**0**answers

44 views

### A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...

**1**

vote

**0**answers

30 views

### How do I find the portion of a cell/voxel lying within a defined surface?

We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$.
A cone-like surface is defined by some function, $z = f(x, y)$, which in ...

**7**

votes

**1**answer

541 views

### Newman's proof of the prime number theorem

I am teaching a graduate course in Complex Analysis and I am covering Newman's proof of the prime number theorem. I have been using the simplified version in the papers of
Zagier and Korevaar. However,...

**1**

vote

**1**answer

59 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

187 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

79 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 $\lambda\gg 1$...

**0**

votes

**0**answers

73 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) ...

**3**

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?
$$
\...

**0**

votes

**0**answers

75 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 $Q_p=\{(x,y)\in\mathbb{R}^2, x>0,y>0,x^p+y^p\leq1\}$, $0<p\leq\infty$ ?
...

**0**

votes

**0**answers

35 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 ...

**2**

votes

**0**answers

159 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)])...

**0**

votes

**0**answers

82 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^\...

**1**

vote

**2**answers

136 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 ...

**-1**

votes

**1**answer

97 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

75 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 $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...