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, ...
1,443
questions
-1
votes
0
answers
44
views
Integrating $\int_{-\infty}^{\infty}\exp(-x^2/(1-ix))(1-ix)^{-1/2}{\rm d}x$
I want to calculate the integral
$$
I=\int_{-\infty}^{\infty}\exp\left(\frac{-x^2}{1-ix}\right)(1-ix)^{-1/2}\ {\rm d}x.
$$
I got two ways to deal this integral. The first, noting that
$$
\int_{-\infty}...
-1
votes
0
answers
29
views
Interchange of supremum and integral for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$
Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere
$$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$
Let ...
-2
votes
0
answers
96
views
Residue theorem and $\pi$ [closed]
I want to prove the following formula:
$$\int_{0}^{+\infty} \frac{\cos(x)}{1+x^2}dx=\frac{\pi}{2e}$$
My question is: is the Residue theorem useful to prove the equality?
15
votes
1
answer
513
views
Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis (comes from a probability question)
This question resisted attacks at MSE, so I am posting it here.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and ...
0
votes
0
answers
74
views
Possible closed form of the integral contains delta function?
I want to evaluate an $N$-dimensional integral defined by
$$
P(s,k)=\int_{-\infty}^{\infty}{\rm d}x_1\dotsi\int_{-\infty}^{\infty}{\rm d}x_N\ \delta\left(\Biggl|\frac{\sum_{m\ne n}^N\exp(i(m+n)k)(x_m-...
0
votes
0
answers
119
views
Help me find the antiderivative of $W(W(x))$ where $W$ denotes the Lambert W Function
Let $W$ denote the Lambert W Function. I must know the antiderivative of $W^2 = W(W(x))$.
I'm already convinced this function is not elementary. This does nothing to settle up my curiosity, as I ...
0
votes
0
answers
25
views
Integral of Cauchy distribution? [migrated]
I'm studying for my midterm and got stuck in a basic probability question. The question is as given below.
Consider Neyman-Pearson criteria for two Cauchy distributions in one dimension
$$p(x|\omega_i)...
3
votes
1
answer
220
views
Expectation on a Polish space
I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...
-1
votes
0
answers
31
views
Solving a ordinary differential equation Problem [closed]
i totally can't figure out how to solve it. can anyone help how to induce the process to answer ?
problem down below
Find the condition of $k$ such that
$y'=\sqrt{4-xy^2}, \quad y(2)=k$
has ...
1
vote
0
answers
61
views
Discretization of oscillating integral
Suppose I am interested in computing
$$
I \equiv \int_0^B dx \, g(x) f(x)
$$
where $B$ is a known upper bound for the integral,
$g(x)$ is a known oscillating function and
$f(x)$ is a smooth function ...
0
votes
1
answer
89
views
Numerical integration with integrable singularity
Suppose I have a numerical estimation of discrete samples of a smooth function $C(t)$ at $t = a, \dots, T = Na$ and I want to (numerically) compute the integral of $f(t) = \frac{C(t)}{\sqrt{t}}$. In ...
-2
votes
0
answers
92
views
What is the value of the contour integral: $\int_{-\infty}^{\infty} \frac{e^{-ipu}}{\cosh^2(u)} \, du$ [closed]
I'm exploring the integral
$$
\int_{-\infty}^{\infty} \frac{e^{-ipu}}{\cosh^2(u)} \, du
$$
in the context of complex analysis, specifically focusing on its evaluation through the residue theorem for ...
4
votes
1
answer
241
views
Approximating a finite sum with an integral
Consider the following sum (with $a$ being a real number and $N$ an even integer)
$$S(a, N) = \sum_{m=1}^{N/2} \frac{4}{N+1} \sin^2\left( \frac{2\pi m}{N+1} \right)\sin \left( 2 a \cos \left( \frac{2\...
2
votes
1
answer
269
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
0
votes
0
answers
69
views
Schauder basis of L^p on general domains
Let $\Omega$ be a non-empty regular compact subset of $\mathbb{R}^n$; i.e. the closure of its interior is itself; for some $n\in \mathbb{Z}^+$. Let $1\le p<\infty$. When does the space $L^p(\...
0
votes
0
answers
91
views
Calculate special integrals [migrated]
Prove:$$\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)$$
I try to use the Taylor expansion of $1-\cos x$ and ...
1
vote
1
answer
95
views
An inequality for the Darboux integral
As we know, if $f(x)$ are Riemann integrable, we have
\begin{gather}
\left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x.
\end{gather}
So, for Darboux integrals, such as the upper ...
1
vote
0
answers
195
views
Prove the limit of the integral [migrated]
Suppose:
$f:[0,\frac{\pi}{2}]\to \mathbb{R}$, $f(x)\in C^2[0,\frac{\pi}{2}]$, $\gamma=\lim_{n\to \infty}(1+1/2+\dotsb+1/n-\ln{n})$.
Prove:$$ \lim_{s\to +\infty}\left(\int_{0}^{\frac{\pi}{2}} \frac{\...
-3
votes
0
answers
133
views
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 ...
6
votes
1
answer
226
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 ...
1
vote
0
answers
129
views
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 ...
1
vote
0
answers
51
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 ...
1
vote
1
answer
69
views
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 ...
0
votes
1
answer
74
views
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 ...
5
votes
0
answers
381
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 ...
3
votes
1
answer
213
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 ...
0
votes
0
answers
106
views
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.
...
1
vote
0
answers
86
views
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 $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
4
votes
2
answers
645
views
Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
Let's state with $\psi^{(1)}$ the trigamma. Calculate the order:
$$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$
(Cornel Ioan Valean)
I uploaded this question here https://math....
7
votes
1
answer
330
views
Exponential trigonometric integral
I want to compute the normalization constant of some probability density on SO(3). After some simplification, I arrive at the following double integral:
$$ \tag{1}\label{eq:1}
\int_0^{2 \pi} \int_0^{\...
2
votes
0
answers
155
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 $...
5
votes
1
answer
316
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 ...
1
vote
0
answers
70
views
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
$$
...
0
votes
1
answer
98
views
Demonstrating bound on integral
I found this question on another forum and it got me interested because of how tight the bound is: prove that
$$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$
The difference ...
1
vote
0
answers
113
views
Integration in polynomial time
The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
7
votes
1
answer
456
views
How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$
$$ \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}^{\...
1
vote
0
answers
54
views
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 ...
2
votes
0
answers
89
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 ...
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 $p\in(-1, 0) \cup (1, \infty)$, ...
0
votes
0
answers
47
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(...
2
votes
1
answer
172
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{\...
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 ...
2
votes
0
answers
155
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 $J_{0}$ ...
7
votes
2
answers
284
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 ...
1
vote
1
answer
90
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 $\...
5
votes
2
answers
273
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, ...
1
vote
1
answer
102
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:
$$...
7
votes
3
answers
875
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 ...
2
votes
0
answers
938
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^...
1
vote
1
answer
204
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 ...