All Questions
Tagged with integral or integration
1,506 questions
1
vote
1
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89
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II
This is a follow-up to this previous question, but under stronger assumptions.
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde ...
- 12.6k
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.6k
4
votes
1
answer
136
views
Decreasing tail integrals for nonnegative random variable $X$
Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also
$$\ell(x):=\frac{1}{...
- 5
2
votes
2
answers
249
views
Assigning values to divergent oscillating integrals
I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ...
- 1,730
2
votes
0
answers
253
views
How to perform this Gaussian matrix integral?
I'm reading this book The supersymmetric method in random matrix theory and applications to QCD. In page 302-303, the author calculate the following integral
$$
Z=\int d\psi \, dH \, P(H)\exp\left(-\...
- 375
1
vote
1
answer
210
views
On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,143
2
votes
1
answer
119
views
Power series expansion of the order parameter in the Kuramoto model
In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,
$$
r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
...
- 640
0
votes
1
answer
73
views
A solution satisfying an integral inequality is bounded [closed]
Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality
\begin{equation}
y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2}
\end{equation}
...
- 11
1
vote
1
answer
177
views
Finding $W^{1,\infty}$ solutions to an integral equation by fixed point
Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation
\begin{align*}
u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s.
\end{...
user139844
2
votes
0
answers
84
views
A closed expression for definite integral of a rational function
Suppose $F(x) = P(x)/Q(x)$ is an integrable rational function on $\mathbb R$, that is, $\deg Q \geq \deg P + 2$, and $Q$ has no real roots.
Does there exist an expression for the definite integral $...
- 481
16
votes
1
answer
2k
views
Equivalence between Lebesgue integrable and Riemann integrable functions
As the title says, for every Lebesgue integrable function $f:\mathbb{R}\to\mathbb{R}$ is there a Riemann integrable function $g:\mathbb{R}\to\mathbb{R}$ such that $f=g$ almost everywhere?
For example, ...
- 285
5
votes
0
answers
198
views
If $\lim_{t\to +\infty} \int_{0}^{\pi} f(x)\exp(e^{xt}) \, dx=0$ then $f=0$ a.e?
Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and
$$
F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt
$$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is ...
- 1,503
1
vote
0
answers
199
views
Integration with respect to Haar measures normalised over a subspace
Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...
- 111
0
votes
1
answer
205
views
Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation
We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are:
$$q_{ij}(t)= \lim_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\...
- 3
2
votes
0
answers
94
views
Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$
I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights,
$$
\int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...
- 21
2
votes
1
answer
309
views
Bounded operator on $L^2(\Bbb R^2)$
Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$:
$$
f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
- 521
0
votes
0
answers
136
views
Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$
where $\...
- 109
2
votes
1
answer
353
views
Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
- 521
0
votes
0
answers
120
views
How to prove an equality involving Laguerre polynomials
Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$.
How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
- 521
4
votes
1
answer
166
views
Estimate an improper integral
Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established?
$$ \...
- 419
8
votes
1
answer
2k
views
How badly can the Lebesgue differentiation theorem fail?
Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that
$$
\lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ?
$$
This is obvious if $0$ is a Lebesgue point ...
- 1,149
1
vote
1
answer
241
views
Integration by parts for indicator of a sphere to indicator of a ball
Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
3
votes
0
answers
159
views
Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?
Background
Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
- 4,829
7
votes
4
answers
1k
views
Integral of product of gaussian CDF and PDF
Looking for an analytic solution to the integral below:
$$
\int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx
$$
where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
- 73
3
votes
0
answers
46
views
Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size
I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\...
- 433
11
votes
1
answer
588
views
Malmsten-like integrals for $\zeta(n) $
Context
Let us define a Malmsten integral when it is of the form $$M_{P,Q} := \int_{0}^{1} \frac{P(x)}{Q(x)} \ln \ln \left( \frac{1}{x} \right) dx. \tag{1} $$ In a paper by Blagouchine [1], the author ...
- 4,829
1
vote
2
answers
167
views
Asymptotic estimation of an integral
I have an integral of the form
$$
I = \int\limits^{1}_{0} \exp\left(\dfrac{vt}{(v+1)^2 + v^2} - vt\right) dv
$$
and I want to prove that $I\leq c t^{-1}$ for the sufficiently large $t$, where $c$ is a ...
- 13
2
votes
0
answers
70
views
How to extend this sum involving generalized harmonic numbers?
It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
- 13.4k
2
votes
1
answer
125
views
For which value of $C(f)$ would the following inequality hold?
I am wondering what would be the value of $C(f)$ for the following inequality to hold? E.g., $C(f)$ could be some quantity related to the Lipschitz constant or the size of the domain.
$$\left(\int f(x,...
- 23
8
votes
0
answers
296
views
Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
- 1,459
3
votes
1
answer
198
views
Volume of 3-dimensional region
Let $G$ be bounded finitely connected domain in $\mathbb{R}^3$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature.
Each sufficiently small open ...
- 197
1
vote
1
answer
173
views
Integral involving Bessel and Laguerre function
Is there a formulas for the following integral
$$\int^\infty_0 e^{-ar^2}L^1_k(b r^2)J_1(cr)r^d dr $$
where $L^1_k$ is the Laguerre polynomials of type 1 and $J_1$ is the Bessel function with $a,...
- 109
-5
votes
1
answer
177
views
Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones? [closed]
Below, we interpret divergent integrals as germs of partial integrals at infinity:
$$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$
where $\operatorname{bigpart}$ means taking ...
- 10.1k
3
votes
0
answers
166
views
An inequality for integrals involving Laguerre polynomials
Let $k\ge n$ and $$A(k,n)=\frac{ \Gamma[1+k]}{n!\Gamma[1+k-n]^2}\int_0^\infty \frac{e^{-r}r^{k-n}}{L_n(-r)} dr$$
where $$L_n(-r) = \sum_{m=0}^n \frac{\Gamma(1+n)}{\Gamma(1+m)^2 \Gamma(1+n-m)}r^m$$ is ...
- 333
3
votes
2
answers
697
views
Class of Riemann integrable functions with antiderivative
We know that continuous functions
are Riemann integrable and
have an antiderivative.
For each bounded function $f$ on an interval $[a, b]$, the Lebesgue integrability theorem guarantees that such an ...
- 263
2
votes
0
answers
136
views
Multiple integral with diagonal constraint (short-range)
I am looking for an upper bound on the following integral:
$$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$
...
- 5,474
3
votes
2
answers
302
views
Definite integral of Bessel function of the first kind times $x^{3/2}$
I am looking for preferably a closed form (or series solution if not possible) for the following integral:
$$\int_0^a x^{3/2} J_\nu (bx) dx$$
where $\nu$ is an integer. This 1D integral appears when ...
- 83
3
votes
1
answer
267
views
References for $\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$ and related integrals?
In user dxdydz's answer to the question "Unexpected appearances of $\pi^{2}/6$", the identity $$\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$$ is mentioned.
...
- 4,829
3
votes
0
answers
214
views
How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
From numerical experiments in Mathematica, I have found the following expression for the integral:
$$
\int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
- 31
2
votes
1
answer
141
views
The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$
Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
- 521
1
vote
1
answer
134
views
Do these two pairs coincide at small time?
Let $\alpha, \beta:\mathbb R_+\to [0,1]$ be continuous and decreasing functions s.t. $\alpha(0)=1=\beta(0)$ and $\alpha, \beta$ are continuously differentiable on $(0,\infty)$ satisfying for some $c&...
- 1,334
2
votes
2
answers
272
views
The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$
How to prove the following inequality $$\forall t>0,\quad\int^\infty_0 \sin(rt)\frac{r^3}{\sinh^2(r)} dr\leq c \big(te^{-At}\big)$$
for some constants $A>0,c>0$
- 521
0
votes
0
answers
83
views
Integral over $S^{n-1}$ [duplicate]
What is the values of the following integral:
$$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw.$$
where $\lambda\in\Bbb R, i^2=-1,x\in\Bbb R^n;<,>$ the inner product scalar on $\Bbb R^n$ ...
- 521
1
vote
0
answers
56
views
Differentiability of functions given as integral of some singular kernel
Let $A: \mathbb R_+\to [0,1]$ be $1/2$-Holder continuous and $k: \{(s,t): 0\le s\le t\}\to\mathbb R$ be continuous. Define $f:\mathbb R_+\to\mathbb R$ by
$$f(t):=\int_0^t\frac{k(s,t)}{\sqrt{t-s}}\big(...
- 1,334
4
votes
1
answer
162
views
Definite integral of power of sine ratio
I stumbled on the following rather appealing trigonometric definite integral,
\begin{equation}
\int_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)}
\end{...
- 3,927
2
votes
1
answer
421
views
Double integral in a polygon domain
I want to compute a integral of a polynomial $f(x, y)$ over a polygon domain $D$ of $n$ sides.
$$
I(f) = \int_{D} f(x, \ y) \ dx \ dy
$$
The vertex of this polygon are
$$\vec{p}_{i} = (x_i, \ y_i) \ \ ...
- 253
6
votes
1
answer
584
views
Integral representation of $\frac{355}{113}-\pi$? [duplicate]
It is well known that
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
Given that $\frac{355}{113}$ is an excellent approximation of $\pi$, is there any known integral representation of ...
- 1,608
0
votes
0
answers
75
views
Dense subspace of square integrable functions on the complex disc
Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...
1
vote
0
answers
141
views
How to find the exact value of $\int_{0}^{\pi} \ln (b \cos x+c)$ without Feynman’s Technique Integration? [closed]
I shall find the integral by Feynman’s Technique Integration on a particular integral
$\displaystyle I(a)=\int_{0}^{\pi} \ln (a \cos x+1) d x,\tag*{} $
where $-1\leq a \leq 1.$
$\displaystyle \begin{...
- 119
0
votes
0
answers
66
views
Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463