# 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, ...

979
questions

**1**

vote

**1**answer

216 views

### Computing the integral $\int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)$

I would like to compute the following integral:
$$
I_\ell(\alpha) := \int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)
\tag{1}
\label{1}
$$
where $\alpha \geq 0$, $J_0$ is the zeroth-order ...

**2**

votes

**0**answers

39 views

### Removing integral from norm by inequality

My first question on Math Overflow.
For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....

**6**

votes

**2**answers

224 views

### An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...

**-2**

votes

**1**answer

190 views

### Is it possible to express $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?

let $\epsilon >0$, I tried to evaluate $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ ...

**5**

votes

**2**answers

418 views

### Compute $ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $ [closed]

How can I compute this integral?
$$
\int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx
$$

**-1**

votes

**1**answer

199 views

### What is the integral of $r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)}$?

I have been trying to solve a research problem for a while now and in doing so, I stumbled upon the following integral:
$$\int_0^{\infty } r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^...

**0**

votes

**1**answer

47 views

### Looking for a family of random variables such that only the second clause is fulfilled [closed]

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if
i) $sup_{i \in I} E(X_i) <\infty$
ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....

**1**

vote

**0**answers

85 views

### A Fredholm equation with non-separable kernel [closed]

I'm trying to solve this form of Fredholm equation:
$$
g(v)=f_1(v)+\int\limits_{0}^{v_\mathrm{th}} g(v_s)\frac{e^{-\tfrac{\big[(v-v_\mathrm{init})-(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\...

**5**

votes

**1**answer

216 views

### Three integral expressions for integer values of $\zeta(s)$. Could these be further reduced to known integrals?

In this MSE-question I've asked about three, similarly shaped, integrals for integer vales of $\zeta(s)$ that I found numerically:
$$\zeta \left( 3 \right) =\frac12{\int_{0}^{1} \frac{1}{x}\big(\zeta(...

**4**

votes

**2**answers

304 views

### From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...

**0**

votes

**0**answers

92 views

### Solving integral equation with an unknown probability distribution

Considering this system of integral equations, where $\gamma \in \mathbb{R} $ and $\alpha\in \mathbb{C}$ are the unknown to solve :
$$ 1=\int_{-\infty}^{\infty} p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\...

**7**

votes

**0**answers

147 views

### A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...

**1**

vote

**0**answers

53 views

### Differentiable dependence on the initial condition of the solution of a SDE

Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...

**4**

votes

**1**answer

292 views

### Two definitions of $L^p$ spaces that are not always equivalent

There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...

**2**

votes

**0**answers

67 views

### Constructing a small Radon-Nikodym derivative

Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that:
$0<h(x)$.
$\int_{x \in \mathbb{R}^n} |h(x)|<\infty$,
$\sup_{x \in ...

**0**

votes

**0**answers

39 views

### Bounding the absolute value of a complex integral with itself

I already asked a similar question on this topic, but after a small discussion, I noted that I did must boil down the problem such that the solution space so to say to maybe have a concrete answer. I ...

**0**

votes

**0**answers

54 views

### Bounding the absolute value of a complex integral

I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function $f(ix)$, with $x\in\mathbb{...

**0**

votes

**1**answer

54 views

### Variant of modified Bessel functions

Consider the integral
\begin{align*}
g_f(x)=\int_{\phi=0}^{2\pi} f(\phi) ~e^{x cos(\phi)}~\mathrm{d}\phi,
\end{align*}
where $f(\phi)$ is a probability density functions defined over $[0,2\pi]$,
\...

**0**

votes

**0**answers

43 views

### Oscillatory integral independent of a parameter

Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral
$$Q(t) :=...

**2**

votes

**1**answer

61 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

222 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(...

**1**

vote

**1**answer

107 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}\...

**-1**

votes

**1**answer

64 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^...

**0**

votes

**0**answers

53 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....

**23**

votes

**1**answer

418 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

74 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

115 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)^\...

**0**

votes

**0**answers

41 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 $C_1$),
\...

**3**

votes

**0**answers

56 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 ...

**2**

votes

**0**answers

33 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

491 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

130 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

47 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 ...

**2**

votes

**1**answer

128 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 ...

**6**

votes

**1**answer

189 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

108 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

77 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

27 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}\}
...

**7**

votes

**2**answers

366 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

78 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

103 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

82 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

89 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

134 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

**1**answer

156 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

31 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

561 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

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 ...