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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Calculate special integrals

Prove:$$\int_0^1\frac{1-\cos x}{x} \, dx-\int_1^{+\infty}\frac{\cos x}{x} \, dx=\gamma, \\ \gamma=\lim_{n\to\infty}(1+1/2+\cdots+1/n-\ln n)$$ I try to use the Taylor expansion of $1-\cos x$ and ...
MathNoob's user avatar
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An inequality for the Darboux integral

As we know, if $f(x)$ are Riemann integrable, we have \begin{gather} \left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x. \end{gather} So, for Darboux integrals, such as the upper ...
Daeree's user avatar
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1 vote
1 answer
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Prove the limit of the integral

Suppose: $f:[0,\frac{\pi}{2}]\to \mathbb{R}$, $f(x)\in C^2[0,\frac{\pi}{2}]$, $\gamma=\lim_{n\to \infty}(1+1/2+\dotsb+1/n-\ln{n})$. Prove:$$ \lim_{s\to +\infty}\left(\int_{0}^{\frac{\pi}{2}} \frac{\...
MathNoob's user avatar
-3 votes
0 answers
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Extending the proof of Maschke's Theorem from finite groups to algebras

In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
Dale's user avatar
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0 answers
80 views

Is it possible to evaluate $ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $ [closed]

People suggested me to upload this question on math overflow and many people gave the numerical results by desmos and wolfram alpha Motivation for the problem here: Is it possible to evaluate $$ \...
Sbsty 's user avatar
  • 111
6 votes
1 answer
198 views

Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here. Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
D.R.'s user avatar
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1 vote
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Can this integral be solved analytically

I have an integral of the form $$\int_{t_1}^{t_2} \frac{\sum_{i=1}^n a_i e^{b_i t}}{\sum_{i=1}^n c_i e^{d_i t}} dt$$ Where $a_i,b_i,c_i,d_i$ are $4n$ real constants, and $t_1,t_2$ are positives. Is ...
lrnv's user avatar
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42 views

Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution): \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} and define its ...
GigaByte123's user avatar
1 vote
1 answer
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Calculating an integral involving Haar measure on orthogonal projections

Let $\sigma_{n, m}$ denote the uniform measure over matrices $U \in \mathbb{R}^{n \times m}$ satisfying $U^T U = I_m$. Let $1_k \in \mathbb{R}^k$ denote the vector with all entries equal to $1$. I am ...
Drew Brady's user avatar
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1 answer
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Uniform convergence of differential quotients in $L^1$

I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
Mario Vasilija's user avatar
5 votes
0 answers
362 views

How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
Jorge Zuniga's user avatar
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3 votes
1 answer
211 views

How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $

I got this general formula for $ n\in N$ (I showed it here) $$\int_0^1 \left(\frac{x}{1-x} \ln x \right)^n dx=n \sum_{p=0}^{n-1}a(n,p+1) (-1)^{n-p} \zeta(p+2)+n! $$ where $a(n,k)$ is the coefficient ...
Faoler's user avatar
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Is there a closed form of $\int\frac{x\ln x}{\arctan x}dx$?

Consider the integral$$\int\dfrac{x\ln x}{\arctan x}dx\label1\tag1$$While I have made some progress on computing the antiderivative, I would like to know whether or not there is a closed form of it. ...
CrSb0001's user avatar
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Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers https://math.stackexchange.com/q/4874446/1298448 How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
Martin.s's user avatar
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4 votes
2 answers
623 views

Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$

Let's state with $\psi^{(1)}$ the trigamma. Calculate the order: $$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$ (Cornel Ioan Valean) I uploaded this question here https://math....
Martin.s's user avatar
  • 194
7 votes
1 answer
327 views

Exponential trigonometric integral

I want to compute the normalization constant of some probability density on SO(3). After some simplification, I arrive at the following double integral: $$ \tag{1}\label{eq:1} \int_0^{2 \pi} \int_0^{\...
Peter Johnson's user avatar
2 votes
0 answers
133 views

Taylor coefficients of the integral of the ordered exponential

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...
Pavel Gubkin's user avatar
5 votes
1 answer
311 views

Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE Certainly, I apologize for any oversight. Here's a more refined ...
Martin.s's user avatar
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1 vote
0 answers
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Integral inequality related to the (mixed?) moments of two functions

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set $$ S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}. $$ Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce $$ ...
Pavel Gubkin's user avatar
0 votes
1 answer
98 views

Demonstrating bound on integral

I found this question on another forum and it got me interested because of how tight the bound is: prove that $$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$ The difference ...
Ivan's user avatar
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1 vote
0 answers
112 views

Integration in polynomial time

The work of Friedman and Ko and Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
poeaqnwgo's user avatar
7 votes
1 answer
453 views

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$ $$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$ where $$ B_n=\sum_{k=0}^{\...
Faoler's user avatar
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1 vote
0 answers
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Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
Didier Felbacq's user avatar
2 votes
0 answers
88 views

Techniques for computing integrals on $G/K$

Edit: I originally tried writing a simpler question, but to make more clear what I want I wrote the general question below Let $G$ be a compact group with compact subgroup $K\subseteq G$ (everything ...
freeRmodule's user avatar
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8 votes
2 answers
2k views

Solving 'impossible' integrals with a new (?) trick

The following identities have been suggested based on formulas in a previous question of mine. If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, ...
Emmanuel José García's user avatar
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0 answers
45 views

Computing the Laplace transform of an expression

I would like to find the Laplace transform of the following expression with respect to the Laplace parameter s $ \int_{z=u}^{\infty} e^{-az/c} g^{'}(\dfrac{z-u}{c}) \int_{x=0}^{\infty} \varphi(z-x)dF(...
Rosy's user avatar
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2 votes
2 answers
159 views

Integral of nested trigonometric function $\frac{\cos\left(p\cos\left(x\right)\right)}{p\cos\left(x\right) + c}$

while still working on my problem $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$ I came across the following definite integral \begin{equation} \int_{0}^{\pi}\frac{\...
Dennis Marx's user avatar
9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
pie's user avatar
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2 votes
0 answers
142 views

Integral of product of zeroth-order bessel functions times cosine $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$

I am new to Bessel functions and need to solve the following integral \begin{equation} \int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x \end{equation} with $J_{0}$ ...
Dennis Marx's user avatar
7 votes
2 answers
282 views

Étendue measure of the set of lines between two Euclidean balls

Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the ...
Gro-Tsen's user avatar
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1 vote
1 answer
82 views

fourth-order multivariate Gaussian integral

I am struggling with an integral of form $$ \int_{\mathbb R^n} y\otimes y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y). $$ I assume that it will involve the trace of some product of $R$ and $\...
Philipp Wacker's user avatar
5 votes
2 answers
262 views

Small parameter expansion of an integral

I am trying analyze an integral of the form $$I(\varepsilon)=\int_0^\infty f(t,\varepsilon) \,dt$$ where $\varepsilon$ is a small real parameter. The function $f(t,\varepsilon)$ is very complicated, ...
Matěj Kudrna's user avatar
1 vote
1 answer
100 views

Analytical solution for a double integral involving logistic functions and Gaussian distributions

I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows: ​$$...
Charles's user avatar
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7 votes
3 answers
869 views

Using the Stone-Weierstrass theorem to solve an integral limit

The following question was posted on math stack exchange here but it got no answers Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
Shthephathord23's user avatar
2 votes
0 answers
936 views

On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality Let $A$ be a non-constant operator acting on $C^...
matilda's user avatar
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1 vote
1 answer
198 views

Antiderivative of Meijer G-function

In the python sympy CAS framework one strategy to compute integrals is to transform the integration kernel to a MeijerG-function, obtain the corresponding antiderivative as MeijerG-function and ...
maliesen's user avatar
  • 284
3 votes
2 answers
525 views

Upper bound for complex integral

I am interested in obtaining a good upper bound for the absolute value of the following integral $$ \left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|, $$ when $n>k>0$ are ...
user512026's user avatar
2 votes
1 answer
123 views

How to calculate this integral of squared Tricomi hypergeometric function

How to solve this integral $$ \int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r $$ where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...
tsukatsuki_sorano's user avatar
3 votes
1 answer
213 views

Hyperelliptic integrals

I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
mxjia's user avatar
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1 vote
0 answers
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$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
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1 vote
1 answer
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The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$

In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...
Ilovemath's user avatar
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-1 votes
1 answer
116 views

A complex complex integral operation [closed]

This question mainly asks about the integral of complex numbers. This question originates from the optical properties of the axicon angle.$\lambda, f,R,ρ_{0},k$ are a constant. $$m(r,\varphi)=\frac{1}{...
Chun Li's user avatar
3 votes
2 answers
600 views

Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
gdoug's user avatar
  • 149
1 vote
1 answer
155 views

Derivation of indefinite integral involving hypergeometric function

I am doing a project on projectile motion and I ended up with this integral: $$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$ where $g, c,$ and $m$...
Leo McIntyre's user avatar
4 votes
2 answers
1k views

Does a function exist which is not Riemann integrable and satisfies the given condition:

I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that $$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...
Lakshmi Priya's user avatar
1 vote
0 answers
60 views

Product of d-dimensional Legendre polynomials

Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
giladude's user avatar
  • 155
5 votes
0 answers
232 views

Is there a way to solve this integral on the sphere explicitly?

Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that $k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral $$f(y):=\int_{\...
Medo's user avatar
  • 698
6 votes
1 answer
367 views

When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals, \begin{align} x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\ y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
Math_Y's user avatar
  • 311
3 votes
1 answer
181 views

If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?

Let $a,b\in\mathbb R$ with $a<b$ and $f:[a,b]\to\mathbb R$. Assume that there exists a Riemann integrable function $g:[a,b]\to\mathbb R$ such that $f=g$ almost everywhere. Then we can NOT conclude ...
Fergns Qian's user avatar
10 votes
1 answer
525 views

Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let $$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$ and $$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$ It ...
Fergns Qian's user avatar

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