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, ...

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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 ...
Wolfgang's user avatar
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2 votes
1 answer
120 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,...
Alec Wang's user avatar
8 votes
0 answers
280 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 ...
TheSimpliFire's user avatar
3 votes
1 answer
196 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 ...
HyyFly's user avatar
  • 187
1 vote
1 answer
152 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,...
Ryo Ken's user avatar
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-2 votes
1 answer
116 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 ...
Anixx's user avatar
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3 votes
0 answers
155 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 ...
MathArt's user avatar
  • 333
3 votes
2 answers
517 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 ...
user7427029's user avatar
2 votes
0 answers
122 views

Multiple integral with diagonal constraint (short-range)

I am looking for an upper bound on the following integral: $$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$ ...
Thomas Kojar's user avatar
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2 votes
2 answers
257 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 ...
Alex's user avatar
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3 votes
1 answer
253 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. ...
Max Muller's user avatar
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3 votes
0 answers
191 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{...
Chris's user avatar
  • 31
2 votes
1 answer
138 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)...
zoran  Vicovic's user avatar
1 vote
1 answer
132 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&...
GJC20's user avatar
  • 1,220
2 votes
2 answers
270 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 $A>0,c>0$
zoran  Vicovic's user avatar
0 votes
0 answers
77 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 $\Bbb R^n$ ...
zoran  Vicovic's user avatar
1 vote
0 answers
53 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(...
GJC20's user avatar
  • 1,220
4 votes
1 answer
154 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{...
Timothy Budd's user avatar
  • 3,545
2 votes
1 answer
335 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) \ \ ...
Carlos Adir's user avatar
6 votes
1 answer
546 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 ...
minhtoan's user avatar
  • 1,404
0 votes
0 answers
65 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-...
Assinisa Hamidata's user avatar
1 vote
0 answers
127 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{...
Lai's user avatar
  • 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 ...
fsp-b's user avatar
  • 421
0 votes
1 answer
121 views

Could variable be still function in x and y after performing Reynolds averaging over area

All, Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$? Equations (1-...
Kernel's user avatar
  • 101
2 votes
0 answers
141 views

Applying 1D integral to matrix integral

In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
Alex's user avatar
  • 73
2 votes
0 answers
127 views

Second differential of total variation

I am trying to give meaning to the notion of second differential of total variation. For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by $$TV(u)=...
Marko Rajkovic's user avatar
1 vote
1 answer
188 views

Expressing the integral over boundary of a domain as an integral over the domain

Let $\Omega \subset \mathbb{R}^2$ be a domain which is "well behaved" (has all "wishable" properties), so as its boundary. For every $u \in C^\infty(\Omega,\mathbb{R})$, I would ...
Marko Rajkovic's user avatar
7 votes
3 answers
640 views

Asymptotics for $\int\exp( -x t / \log t)dt$

What is the asymptotic growth rate of $$f(x) = \int_e^\infty e^{ - x t / \log t} dt$$ as $x \to 0$? As an example of what is meant by "growth rate" consider $$g(x) = \int_e^\infty e^{-x t} ...
Matthew Junge's user avatar
3 votes
2 answers
84 views

Properties of functions provided that the integral equation $\int_x^1 a(y-x)a(y) \, dy = \int_x^1 b(y-x)b(y) \, dy$ holds for $x \in [0,1]$

Let $a,b : [0,1] \to \mathbb R$ are two functions (e.g. suppose that they are in $L^2[0,1]$ or are $N$-times continuously differentiable). Now suppose that $$ \int_x^1 a(y-x)a(y) \, dy = \int_x^1 b(y-...
J. Swail's user avatar
  • 347
1 vote
0 answers
66 views

Properties regarding Poisson non-integrability [closed]

Let $a>0$ and $f:[a,\infty]\to [0,\infty)$ be a continuous increasing function. We call $f$ to be "Poisson non-integrable" if $f$ satisfies $$\int_a^\infty \frac{f(x)}{x^2}dx=\infty.$$ ...
user483450's user avatar
2 votes
1 answer
305 views

Evaluation of Gaussian multivariable integral

In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears: \begin{equation} \int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
Alex's user avatar
  • 73
7 votes
2 answers
1k views

How would you work out this integral as a series?

The integral is: $$f(a) = \int\limits_{-\infty}^\infty \frac{x e^{-a^2 x^2}}{\tanh(x)}dx$$ which seems to converge for all $a>0$. But I don't know how to get a sense of the function $f(a)$ such as ...
zooby's user avatar
  • 255
0 votes
1 answer
109 views

upper bound for infimum of integral

I was reading this post , where the following question is discussed: Let $h:[0,1] \rightarrow[0,1]$ be a $C^{1}$ function such that $h^{\prime}(x)<0$ for all $x \in(0,1)$. Then, $$ \inf _{f \in \...
Gordafarid's user avatar
3 votes
1 answer
93 views

On an integral of Gaussian CDFs

Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by $$F(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$ For every $a\in [0,1)$, ...
GJC20's user avatar
  • 1,220
4 votes
1 answer
310 views

Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras. Suppose that I want to define a direct integral on either Banach spaces or ...
Frederik Ravn Klausen's user avatar
4 votes
1 answer
261 views

Mikusiński's approach to Bochner integrals; replace absolute by unconditional?

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions: Defn. Let $X$ be a Banach space. ...
Willie Wong's user avatar
  • 37.4k
1 vote
3 answers
486 views

Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
Alex M.'s user avatar
  • 5,207
0 votes
0 answers
140 views

Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms

Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
Astroichthys's user avatar
1 vote
2 answers
152 views

Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?

Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral. $$ I = \int_{\mathbb{R}} \int_{\mathbb{...
Johnny T.'s user avatar
  • 3,547
2 votes
1 answer
261 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
etal's user avatar
  • 162
1 vote
1 answer
61 views

Weak lower semicontinuity of a sequence of Riemann sums

Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums $$R^K=\frac{1}{K} \sum_{k=0}^{...
Marko Rajkovic's user avatar
0 votes
0 answers
63 views

Integration of matrix form of Vasicek variance (Python/Matlab)

$X_t$ is a vector and follows the following Vasicek process. $$ dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\ $$ What is the variance of $X_t$? In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
JH Y's user avatar
  • 1
1 vote
0 answers
35 views

How to relate this integration with the integral expansion of special functions?

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, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
Rekha K.'s user avatar
1 vote
1 answer
138 views

Can the integral inherit the Lipschitz continuity of its integrand?

Let $C$ be the set of continuous functions on $[0,T]$ taking values in $[0,1]$. Denote $\|f-g\|_t:=\max_{0\le u\le t}|f(u)-g(u)|$ for $f,g \in C$ and $t\in [0, T]$. Let $\phi: C\times C\times \{(s,t): ...
GJC20's user avatar
  • 1,220
1 vote
0 answers
76 views

Integral of $f \in L^{\infty}(\mathbb{T}^2)$

Let $f \in L^{\infty}(\mathbb{T}^2)$, where $\mathbb{T}$ is the torus. Can we somehow compare the integral $$\int_{\mathbb{T}^2}\int_{\mathbb{T}^2} \left( \int_{3x+y=z} f(w+x)f(w+y)dm_{\mathbb{T}^2}(x,...
User's user avatar
  • 195
5 votes
2 answers
245 views

Inverse Mellin transform of 3 gamma functions product

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
Pouya's user avatar
  • 59
0 votes
0 answers
47 views

How was this heat semigroup estimate made in a paper on reaction–diffusion systems?

In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage $$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$ Here $S(t)$ denotes the heat ...
Ilovemath's user avatar
  • 585
2 votes
0 answers
109 views

What is the justification for using Wiener integrals to integrate over a space of differentiable functions?

In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
Harmenszoon's user avatar
3 votes
0 answers
238 views

Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure

Let $(X, | \cdot |)$ be a Banach space. I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\...
ViktorStein's user avatar
3 votes
1 answer
343 views

A specific integration with Grassmann variables

I have recently read (for example, here) that this relation below is true $$ \int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}), $$ where $Pf(\mathbf{A})$ is the Pfaffian of an even ...
251's user avatar
  • 31

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