All Questions
Tagged with integral or integration
1,506 questions
1
vote
1
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119
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Volume of a frustum knowing the volume and height of the pyramid and the height of the frustum [closed]
Can I calculate the volume of a frustum if all I know is the volume of the pyramid the height of the pyramid and the height of the frustum?
- 113
1
vote
0
answers
78
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Is the set of functions between two arbitrarily close L^1 functions closed?
Let $(X, {\cal \tau}, \mu)$ be a probability space
and $F$ some vector subspace of integrable functions (defined everywhere). I can look at the set
$$\hat{F} = \{f \hbox{ integrable } \mid \forall \...
- 18.7k
0
votes
0
answers
103
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Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$
I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral:
$$\frac{1}{\pi}...
- 9
1
vote
1
answer
103
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How to prove that $\int (1-z)^{u} z^{v} dz$ is equal to $\frac{z^{v+1}}{v+1}_2F_1(-u, v+1; v+2; z)$?
How to prove that
$$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, _2F_1(-u, v+1; v+2; z)?$$
1
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1
answer
164
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Two trigonometric integrals: looking for a transformation
I have two integrals of trigonometric functions and I would like to ask:
QUESTION. Is there a transformation rule (or general principle) to show this equality?
$$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
- 43.2k
0
votes
1
answer
243
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Integral form of expectation with respect to complex random variables [closed]
Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
- 13
0
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1
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142
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What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$? [closed]
Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$:
$$\int{x^{-a} (b -cx^{-d})^e dx} = -\frac{b^{e}x^{1-a} \, _2F_1\left(\frac{a-1}{d},-e;\frac{a+d-1}{d};\frac{c x^{-d}}{b}\...
4
votes
1
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114
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Antiderivative of $f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$
I need to know the primitive function (Antiderivative) of this function:
$$f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$$
where
$K$ is an integration constant,
$d=-\frac{1}{2p}$ with $p<...
- 51
0
votes
1
answer
89
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Solution or approximation to $\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx$?
I'm looking for a closed solution or an approximation to $$\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx,$$
where $a, b, c, d > 0$.
1
vote
1
answer
110
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Solution to $\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx$
I'm looking for a solution to $$\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx.$$
Mathematica gives me the following solution, but I'd like to know/understand the steps involved in finding it.
$$\int ...
0
votes
0
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76
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Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$
How could one calculate the closed form solution of this integral:
$\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$
Here the integral is ...
- 71
1
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0
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210
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Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
6
votes
1
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193
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Oscillatory integrals with a decaying factor in the integrand
Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased):
Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
- 63
2
votes
0
answers
86
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The square-integrability of $p$ and $\nabla u$
We consider the stationary Stokes problem in $\mathbb{R}^n$
$$\DeclareMathOperator{\Dvg}{\nabla\cdot}
\begin{cases} \Delta u + \nabla p = f & \text{ in $\mathbb{R}^n$} \\
\Dvg u =0.
\end{cases}
$...
3
votes
1
answer
413
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Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$
$\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with ...
- 5,405
5
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2
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434
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How to integrate the multinomial over a ball in $\mathbb{R}^{n}$?
I got an interesting question. Consider this integral:
$$ \int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x, \quad m,n\in \mathbb{N}, \ a_{i}>0, \ i=1,2,\ldots,n.$$
It is clear that ...
- 51
5
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2
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249
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Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals?
Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g.
$$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$...
- 319
0
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1
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102
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Gronwall-type inequality with nonlinearity
Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$...
user139844
1
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1
answer
613
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Integral of the product of a gaussian pdf and cdf
I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
- 11
6
votes
1
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436
views
Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$
Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers.
\begin{equation}
(\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2
...
- 545
5
votes
1
answer
344
views
Monotonic dependence on an angle of an integral over the $n$-sphere
Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...
- 115
5
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0
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652
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
- 790
1
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0
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99
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Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
3
votes
1
answer
324
views
Does the Radon-Nikodym derivative commute with integration?
Suppose I have a measurable space $(\Omega, \Sigma)$ and a function $f: \mathbb{R} \times \mathbb{\Sigma} \rightarrow [0,1]$ such that for any $x \in \mathbb{R}$ the tuple $(\Omega, \Sigma, f(x, \_))$ ...
- 133
1
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1
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132
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Is there a solution to $\int_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)dx$?
I'm looking for a solution to the following integral.
$$\int_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)dx,$$
where $b,c,d,e> 0$ and $0< a < \lambda < y$.
This equation ...
0
votes
0
answers
54
views
Interchanged Domain of multiple integral under n-simplex
If this question is too elementary, I apologize to ask here.
Suppose that we have a multiple integral under $n$-dimensional simplex as follows:
\begin{align*}
\underbrace{\int_0^1 dx_n \int_0^{x_n} ...
- 284
1
vote
1
answer
441
views
Expectation of the maximum of a lognormal distributed variable and zero
I need to find an algebraic expression for E(max{X-a,0}), where X has a lognormal distribution with mean mu and standard deviation sigma. So far, I have derived the following expression, but I could ...
1
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1
answer
180
views
Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?
I'm looking for a solution to the following integral. However, it seems it doesn't have a solution.
$$\int\limits_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{\displaystyle1}{\...
2
votes
0
answers
168
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Geometric sets determined by chains (for integration and Stokes' theorem)
I have asked a similar question on mathSE more than a year ago, which received no answers, only a few comments which did not really help me. I am now re-asking this question here but reformulated ...
- 3,292
2
votes
1
answer
156
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Given an increasing function $f$, to find a continuous function satisfying properties of $f$
Let $f:[0,\infty)\to [0,\infty)$ be an increasing function satisfying
$$\int_0^\infty f(x)\frac{dx}{1+x^2}=\infty.$$
Can we find a continuous increasing function $F$ on $[0,\infty)$ satisfying
$$\...
- 99
1
vote
1
answer
241
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Integrals of the type $\delta(g^{n})$ on $\mathrm{SU}(2)$
I posted this question previously to MathSE. However, I have still not solved it, so lets try to ask it here. When doing some calculations with spin-foam models for 3d quantum gravity for some ...
- 1,171
6
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0
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249
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Double integral $\int \int (\log x) (\log y)/F(x,y) dx dy$: elegant way?
I need to evaluate (or, if that is not feasible, bound well) some integrals of the type
$$\mathop{\int \int}_{(x,y)\in U} \frac{\log x \log y}{F(x,y)} dx dy,$$
where $U = \{(x,y)\in [1,\infty)^2: F(x,...
- 20.2k
3
votes
1
answer
537
views
Integration of $\int\limits_0^{2\pi} \int\limits_0^{2\pi} \min \{ x-y, 2\pi- (x-y) \} e^{a\cos(x)} e^{a\cos(y)} \, dx \, dy$
$$\int\limits_0^{2\pi} \int\limits_0^{2\pi} \min \{ x-y, 2\pi- (x-y) \} e^{a\cos(x)} e^{a\cos(y)} \, dx \, dy, \qquad a\in\mathbb R.
$$
I tried to find the value of the integral following the method ...
- 41
-2
votes
1
answer
100
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Understanding the performed change of variable in this integration [closed]
I'm stuck on a passage I do not understand, which reads:
$$\int_{r<|y|<1} \bigg| \frac{1}{(|y|^2 - r^2)^s |y|^n} - \frac{1}{|y|^{n+2s}}\bigg|\ \text{d}y$$
$$\int_1^r \bigg| \frac{1}{(t^2 - r^2)^...
- 99
2
votes
1
answer
317
views
Are there any published studies on cases of infinite sums for which the Euler–Maclaurin summation method yields the exact evaluation?
The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(...
- 4,829
2
votes
1
answer
215
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A 2 dimensional integral in polar coordinate [closed]
Recently I got stuck on a 2 dimensional integral in polar coordinate,
the expression is the following:
$I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...
- 71
7
votes
2
answers
2k
views
The source of the Integral
Wolfram alpha calculates the integral
$$\int\limits_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$
However, I need to cite the source of this identity (the table of ...
- 16.5k
6
votes
0
answers
292
views
A kind of reflection formula for the logarithmic derivative of the zeta function
So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
- 13.4k
4
votes
0
answers
249
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A tricky integral equation
In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.)
Let $f:\...
- 9,720
2
votes
1
answer
276
views
Can $\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx $ be evaluated?
Currently, I'm working on a problem pertaining to certain integrals involving the modified Bessel function of the first kind. On p. 59 of this book by Rosenheinrich, it is stated that
$$\int e^{-x} I_{...
- 4,829
3
votes
1
answer
293
views
Derivative of an integral of a Gaussian
I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian:
$ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
- 43
3
votes
1
answer
190
views
Volumes of unit balls in $S^2 \times\mathbb R$ and $H^2 \times\mathbb R$
The following integrals are equal to the volume of a unit ball in $S^2 \times \mathbb R$ and $H^2 \times\mathbb R$, respectively:
$$8\pi\int_0^1\sin^2 \frac{\sqrt{1-h^2}}2 \, dh$$
$$8\pi\int_0^1\sinh^...
- 2,773
5
votes
1
answer
226
views
When do volumes depend real-analytically on the parameters defining the regions?
Suppose $f_1, \dots, f_n$ are real-valued real analytic functions defined on an open set $B=(0,1)^d$ of $\mathbb{R}^d$.
For $r \in \mathbb{R}$, let $S_r$ be the sub-level set in $B$ defined by the ...
- 6,224
3
votes
1
answer
327
views
Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$?
Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?
- 930
1
vote
1
answer
105
views
Integral of $J_1\left(Ae^{-\lambda t}-Ae^{-\lambda s}\right)e^{-\epsilon(t-s)}$ with respect to $s$?
Consider the integral
$$\mathcal{I}=\int_0^t\left(\frac{Ae^{-\lambda t}-Ae^{-\lambda s}}{2}\right)^{2m+1}e^{-\epsilon(t-s)}ds,\tag{1}$$
for constants $A,\lambda,\epsilon,t\in\mathbb{R}$ and $m\in\...
- 79
7
votes
1
answer
293
views
On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series
The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series:
\begin{align*}
\int_{-\infty}^{\...
- 7,609
3
votes
0
answers
397
views
Differential of exponential map with respect to the base point
Let $(M,g)$ be a smooth Riemannian manifold embedded in $\mathbb{R}^m$. I would like to understand the transformation formula which will allow me to pass from the integral $\int_M \dots dV_g(x)$ to $\...
- 319
0
votes
1
answer
167
views
Solution of this differential equation [closed]
I wonder if it is possible to solve analytically the following equation
$$
\dot{\alpha}_t = -\frac{2}{m} \alpha^2_t + \frac{1}{2m} (\alpha_t - \alpha_t^*)^2
$$
Where $\alpha_t$ is a complex function, $...
1
vote
0
answers
124
views
Is it possible to define a Bochner integral for a $S'(\mathbb R^d)$-valued function?
I apologize in advance for the rather vague question.
While reading the book White noise distribution theory by H.H. Kuo, in particular the section 13.3 I came across the following statement (I'll ...
- 515
2
votes
0
answers
104
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Coercivity of an integral operator in control theory
Let us consider the integral operator $T:\mathbb{R}^{n\times d}\to [0,\infty)$ such that for all $K\in \mathbb{R}^{n\times d}$,
$$
T(K)=\int_0^1 \operatorname{tr}(KK^\top \Sigma_t) \,d t,
$$
where $\...
- 503