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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6 votes
3 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 $\phi(\cdot)$ ...
3 votes
0 answers
44 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 $$\...
2 votes
0 answers
69 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 ...
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,...
1 vote
2 answers
482 views

Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain $...
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 ...
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) \ \ ...
-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 ...
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 ...
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 ...
2 votes
0 answers
122 views

Multiple integral with diagonal constraint (short-range)

I am looking for an upper bound on the following integral: $$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$ ...
3 votes
1 answer
357 views

Ability to have function sequence converging to zero at some points

Consider the continuous and non negative function $c : \mathbb R \to [0,1]$ defined by $$ c(x) = \begin{cases} \cos \frac{\pi x}{2} &\text{for } x \in [-1,1]\\ 0 &\text{otherwise} \end{cases}$$...
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. ...
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-...
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)...
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&...
31 votes
4 answers
17k views

About the Riemann integrability of composite functions

When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions. For the composite function $f \circ g$, He presented three cases: 1) ...
24 votes
1 answer
2k views

Why these surprising proportionalities of integrals involving odd zeta values?

Inspired by the well known $$\int_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given here (writing $\zeta_r:=\zeta(r)$ for easier reading)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
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 $A>0,c>0$
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 $\Bbb R^n$ ...
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(...
16 votes
7 answers
6k views

Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...
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 ...
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 \...
3 votes
1 answer
780 views

Integration by parts on manifold with corners

Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e. \begin{align*} \int_M g(\nabla ...
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-...
0 votes
1 answer
94 views

Gronwall-type inequality with nonlinearity

Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$...
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{...
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 ...
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 ...
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)=...
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 ...
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 ...
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-...
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 $$\int_a^\infty \frac{f(x)}{x^2}dx=\infty.$$ ...
4 votes
1 answer
1k views

Normal multivariate orthant probabilities

(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.) Let $\mathbf{...
3 votes
4 answers
820 views

Compute the two-fold partial integral, where the three-fold full integral is known

I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function \begin{equation} 4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1} \rho_{22}^{3 \...
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 ...
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 ...
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. ...
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 $a\in [0,1)$, ...
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, \...
6 votes
4 answers
530 views

Generalizing contour integration to quaternions and bicomplex numbers

I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
2 votes
4 answers
933 views

Change of variables in a Gaussian integral in matrix form

I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$ where ...
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 ...
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 $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
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{...
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}^{...
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}\...
13 votes
2 answers
721 views

How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?

For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...

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