All Questions
Tagged with integral or integration
1,507 questions
5
votes
1
answer
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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{...
- 733
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
0
votes
1
answer
165
views
An extension of the mean-value theorem for integrals? [closed]
The mean value theorem for integrals states that if $f$ and $g$ are continuous on $[a, b]$ and $g$ never changes sign on $[a, b]$, then there exists some $c\in [a, b]$ such that
$$\int_{a}^{b} f(x)g(...
user140392
2
votes
0
answers
197
views
Orthogonality relation in $L^2$ implying periodicity
Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...
- 405
-1
votes
1
answer
126
views
Is there a name for this family of integral?
This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.
- 101
2
votes
0
answers
101
views
exact form of the integral of x^(-x) between 0 and infinity [closed]
I have searched the internet for an exact answer and although I have found many decimal approximations, https://www.wolframalpha.com/input/?i=integrate+x%5E(-x)+from+0+to+infinity I have not been able ...
- 31
0
votes
0
answers
42
views
Specify modified error function in form of error functions
How can we express $\mathrm{erf}(\frac{t-a}{m})$ as a sum of functions of the form $\mathrm{erf}(t)$?
I am developing a fitting routine and I encounter this integral:
$$\int_{0}^{x}\mathrm{erf}\left(\...
-1
votes
1
answer
154
views
About a multiple integral [closed]
In my current research, I'm confronted with the justification of some facts, and I don't know how to proceed in proving them, so I need to know if there exist some theorems (precisely three theorems) ...
- 417
3
votes
0
answers
160
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Elements of vector-valued $L^1$-spaces
Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$...
- 799
2
votes
0
answers
425
views
Why is the integral of the tautological 1-form equal to the action?
I am having a hard time to understand why the integral of the tautological 1-form is the action of the system.
The tautological one form is defined by :
\begin{align}
\theta_{(q,p)} : T_{(q,p)}T^*Q &...
- 71
3
votes
1
answer
986
views
Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual ...
- 315
3
votes
2
answers
304
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The integrals of things looking like $e^{(\frac{a}{z}+\frac{b}{z-c})}$ on closed contours
I have recently encountered a truely terrible integral which I need to compute. I am not sure it's doable but before throwing the whole project in the bin I thought I would ask here. At the moment, a ...
- 989
42
votes
0
answers
2k
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Are we better in computing integrals than mathematicians of 19th century?
When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
3
votes
2
answers
235
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A reduction problem from $\mathbb{R}^2$ to $\mathbb{R}$
Let $f,g \in L^1_\text{loc}(\mathbb{R})$, with $g \geq 0$, and such that for almost every $(x,y) \in \mathbb{R}^2$, at least one of the following equations is true :
\begin{align*}
f(x) + f(y) + g(...
user140442
4
votes
2
answers
718
views
Integrate $1/(x_1x_2\cdots x_n)^k$ for $1\le x_i \le a$, where product of coordinates satisfies $ b\le x_1\cdots x_n\le c$
I need to integrate
$$
\int\limits_{[1,a]^n} \frac {\chi(\{ b \le x_1 \cdots x_n \le c \})} {( x_1 x_2 \cdots x_n)^k} \,dx_1 \cdots dx_n,
$$ where $\chi(E)$ is the characteristic function of a set $E$....
- 91
1
vote
0
answers
146
views
Differential equation with Fresnel integral
We have $\frac{y'(x)}{\cos(x)}=C(x)$ and need to find y(x). Generally we should express $y(x)$ through $C(x)$ and elementary functions. I can only do it through $C(x)$ and $S(x)$, or through $\Phi(x)$....
- 101
1
vote
1
answer
166
views
How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?
To begin, we have two constraints
\begin{equation}
C1=x>0\land z>0\land y>0\land x+2 y+3 z<1
\end{equation}
and
\begin{equation}
C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...
- 723
1
vote
0
answers
123
views
Is this integral zero?
I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation.
Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
- 327
3
votes
1
answer
848
views
Integration by parts on manifold with corners
Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e.
\begin{align*}
\int_M g(\nabla ...
- 156
1
vote
1
answer
814
views
$L^2$ norm of fractional Laplacian
Is it possible to calculate the $L^2$ norm associated to fractional Laplacian of $u$ and $s\in (0, 1).$
$$\|(-\Delta)^{s/2} u\|_2^2=\int_{\mathbb R^N}|(-\Delta)^{s/2} u|^2dx=C_{N,s}\int_{\mathbb R^{...
- 407
2
votes
0
answers
149
views
A bound using Cauchy formula
Let $0<t_0<1$ fixed number , $ n_0$ integer $ \geq 2$ fixed and let $\forall 0<u<1, f(u)= \displaystyle \frac{(1-u)^{n_0} \log(1-u)}{(1-ut_0)^{n_0+1}} $.
Let $0<u_0<1 $ be given. ...
- 417
0
votes
0
answers
60
views
Integral involving legendre (as Beukers integral) [duplicate]
let $\forall n $ integer $p_n(t)=\frac{1}{n!}(t^n(1-t)^n)^{(n)} $
i 'm looking for an explicit constant $0<c<1$ ( very good small c ) independant of $n$, and a constant $b$ ( non explicit) ...
- 417
3
votes
1
answer
244
views
Looking for bound in integral involving Legendre polynomial
I'm looking for an upper bound to the following integral or equivalent when $n$ leads to $ +\infty $ to the following expression
$$I_n:=\left|\int_{0}^1 \int_{0}^1 \frac{p_n(x) p_n(y)}{(1-xy)} dx dy ...
- 417
5
votes
4
answers
953
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
7
votes
3
answers
730
views
$\int_{-\infty}^\infty \frac{e^{-y^2/2}}{((y+y_0)^2+x_0^2)^r} dy$
I have to estimate the integral $$\int_{-\infty}^\infty \frac{e^{-y^2/2}}{((y+y_0)^2+x_0^2)^r} dy,$$ for $r\in \mathbb{R}^+$. I am a little amazed that Sage and Wolfram Alpha have nothing to say about ...
- 20.2k
11
votes
3
answers
861
views
Nonnegativity of an integral over the unitary group
For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
- 1,593
1
vote
1
answer
126
views
The expectation of binary logistics regression with respect to Gaussian distribution
I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If ...
- 37
3
votes
1
answer
223
views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
2
votes
1
answer
83
views
Integral substitution involving the length and angle of two vectors
Let $F\colon\mathbb R^3\to\mathbb R$ be a compactly supported smooth function. I want to compute
$$ \int_{\mathbb R^n}\int_{\mathbb R^n} F(\lVert x\rVert^2,\lVert y\rVert^2,\langle x,y\rangle)~\...
- 623
24
votes
1
answer
2k
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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)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
- 13.4k
5
votes
1
answer
997
views
Double integral with logarithms [closed]
Faster method to calculate the exact solution of the following integral (based on the ideas of Fedor Petrov: https://mathoverflow.net/users/4312/fedor-petrov, Double integral with logarithms, URL (...
1
vote
1
answer
345
views
Identity involving dot product of solid angle and gradient [closed]
How to prove following for $n\geq0$ ?
$$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$
Where, at any point $\vec{r}$, the $\vec{\Omega}$...
- 31
16
votes
0
answers
556
views
Reference request for Grothendieck's work on "Integration with values in a topological group"
Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
user57432
3
votes
1
answer
363
views
Oscillatory integrals
Consider the integrals
$$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$
I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...
- 4,115
1
vote
1
answer
245
views
Evaluation of a double definite integral with a singularity
How to compute the
$$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy.$$
Is it possible to compute the integral analytically upto some terms. I believe it should involve ...
- 407
2
votes
1
answer
803
views
On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
user85801
4
votes
0
answers
192
views
Can this integral be made nonpositive?
Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...
- 2,095
5
votes
1
answer
294
views
Asymptotic of integral $\int_{1}^{e^n}(1-\frac{\ln(x)}{n})^n\,dx$
How could we find the large-$n$ asymptotic of $$\int_{1}^{e^n}\left(1-\frac{\ln x}{n}\right)^n\,dx.$$
I have a suspicion that this is $\sqrt{n}$.
0
votes
0
answers
43
views
integrating multivariable rational function over a product of disks
Suppose I have a rational function of $k$ complex variables:
$$ R(x_1,...,x_k) = P(x_1,...,x_k)/Q(x_1,...,x_k) $$
where $P$ and $Q$ are polynomials. Now I'd like to compute the integral of this ...
- 169
9
votes
1
answer
1k
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Integration by parts formula for the double Riemann-Stieltjes integral
In my research the following integration by parts formula for the double Riemann-Stieltjes integral
$$\int\limits_{[a,b]\times[c,d]}f(x,y)\,dg(x,y)=f(b,d)g(b,d)-f(a,d)g(a,d)-f(b,c)g(b,c)+f(a,c)g(a,c)...
- 3,486
0
votes
0
answers
124
views
Integrating over some domain of the Stiefel manifold to analyze its support
Define the square $d$ dimensional Stiefel manifold as
$$V_{d} = \{ R \in \mathbb{R}^{d \times d} : R ^\top R = I_d \} .$$
How does one integrate on this manifold over a domain defined as $\{ R \in V_{...
- 21
3
votes
0
answers
132
views
A new characterization of Riemann-Integrability
Question :
Given two bounded functions $\,f:[a,b]→\mathbb{R}\,$
and $\;θ:(0,b−a]→[0,1]$.
Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $\,[a,b]$.
Let $\,Δx_k=x_k−x_{k−1}\,$ and $\,\...
- 31
1
vote
1
answer
200
views
About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$
An integral has been pushed me over the edge for several weeks. It reads as:
$$\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$
I tried ...
- 21
17
votes
5
answers
1k
views
Closed-form expression for certain product
$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
- 283
0
votes
1
answer
459
views
A pair of integrals involving square roots and inverse trigonometric functions over the unit disk
I would like to perform the integration ($u \in [0,1]$),
\begin{equation}
\int_{a=-1}^1 \int_{b=-\sqrt{1-a^2}}^{\sqrt{1-a^2}} u^2 \sqrt{-a^2 u^2-b^2+1} \tan ^{-1}\left(\frac{\left| a\right|
}{\...
- 723
5
votes
1
answer
163
views
A question about integration of spherical harmonics on $(S ^ 2, can)$
Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that
$$ \int_{\mathbb{...
- 245
3
votes
0
answers
229
views
Reference for calculating definite integral involving sines
Recently I accidentally discovered a simple, elementary derivation of the following identity, valid for any $n,k \in \mathbb N$:
\begin{align*}
\frac1\pi \int_0^\pi {\rm d}x \left(\!\frac{\sin nx}{\...
- 686
1
vote
2
answers
819
views
Integral formula involving Legendre polynomial
I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...
- 31
6
votes
3
answers
554
views
Computing the volume of a simplex-like object with constraints
For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...
- 61
3
votes
0
answers
426
views
Integration over a Surface without using Partition of Unity
Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
- 277