The integration tag has no wiki summary.

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### Integral of Weingarten Map / Shape Operator [on hold]

This Paper states that the Weingarten Map / The Shape operator $W_p$ of a two-dimensional surface $S\subset\mathbb{R}^3$ at a point $p$ can be expressed in the following way:
...

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41 views

### Integrate Faddeeva function

I came across this integration in my studies.
$\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$
It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...

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100 views

### How to solve definite integral involving exponential function

I am trying to get a closed form for the following definite integral:
$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 ...

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**1**answer

129 views

### Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which
$$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$
and
$$x_1+\ldots+x_k\leq 1.$$
I am ...

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118 views

### Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...

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144 views

### An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j},~
1\leq j \leq 2^{n} \}
\right\} \right)_{n ...

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127 views

### Finding closed form of : $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

$ \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\dd}{{\rm d}}$
Finding closed form of the below:
$$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
This ...

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29 views

### Saddle point method for asymptotic expansion [migrated]

Any idea on how to derive an asymptotic expansion of the following integral for $x$ around zero (using Saddle point method):
$$\int_{0}^{x}\left(u+\alpha\right)^{-a - \left(b-a\right)e^{-\lambda ...

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62 views

### Integration of the reciprocal of sum exponential [migrated]

Any one know the method to do the integration as
$$\int\frac{x^2\cdot \exp (-ax^2) \exp(-bx^2)}{\exp(-ax^2)+\exp(-bx^2)}dx$$
It can be simplified as
$$\int\frac{x^2}{\exp(ax^2)+\exp(bx^2)}dx$$
...

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79 views

### Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...

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vote

**1**answer

69 views

### Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge:
$$
\int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ?
$$
Among special cases are such ...

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153 views

### Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...

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**0**answers

77 views

### motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...

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104 views

### How to evaluate the following integral related to exponential distribution

I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that
$$
...

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63 views

### Equivalence of ensembles and $\sum_{N=0}^{\infty} \int_{-Bn}^{\infty} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $

I am thinking about a step in the proof of the equivalence of ensembles in StatMech on page 25. step from 3.19 to 3.20.
It seems to be argued there that for $A$ and $K$ large enough, the term
...

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votes

**1**answer

100 views

### The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the ...

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80 views

### Reference: Bochner Integral`

What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...

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35 views

### Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as
...

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49 views

### How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...

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34 views

### A hyperbolic partial differential equation

How solve this equation (numeral or analytical)?
$u(t,x)=\int_{t-x}^{t}{a \cdot e^{b \cdot s} \cdot \int_{0}^{s-(t-x)}{u(s,z)dz}ds}+\int_{t-x}^{t}{c \cdot e^{d \cdot s} \cdot ...

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62 views

### How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...

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**3**answers

276 views

### Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath

I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+):
$$
\int_{-1}^1\textrm{d}t ...

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**4**answers

199 views

### Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here.
I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...

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74 views

### Integration by parts for multidimensional Lebesgue-Stieltjes Integrals

I am concerned with the following problem:
I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given ...

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217 views

### Morphisms associated to measured spaces [duplicate]

In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...

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57 views

### L2 norm of a M-Whittaker function

Let $M_{\kappa,\mu}(z)$ be the Whittaker function, as defined here http://en.wikipedia.org/wiki/Whittaker_function.
Does any one know the evaluation of the following integral?
...

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**1**answer

95 views

### Behavior of the integral of products of probability densities

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := ...

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29 views

### Integrating over the Intersection of Convex Regions

Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?
The ...

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31 views

### Solution of ODE related to normal density

This question below is from mathse. I reqeustioned here because very limited respond there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...

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73 views

### Non Borel Spaces: Gauge Integral

Question
Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology?
I'm wondering since it seems as the gauge crucially uses ...

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**1**answer

139 views

### Contour integral around semi-circle

Can one use contour integration to evaluate $\int^{\pi}_{0} \frac{1}{1-\rho*sin(\theta)}d\theta$ for $0<\rho<1$? This would be trivial if the upper limit were $2\pi$ as we could let ...

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**1**answer

200 views

### An inequality concerning convexity and expectation

Assume $f$ and $g$ are nonnegative with
$$\int_0^\infty f(x)dx=1=\int_0^\infty g(x)dx
$$
and
$$\int_0^\infty xf(x)dx<\infty > \int_0^\infty xg(x)dx
$$
Is it true for nonnegative numbers $p$, $q$ ...

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93 views

### Question about a particular estimate in Riemannian geometry

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...

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**2**answers

121 views

### Evaluate an integral or Fourier coefficients

Consider an integral
$$
\int_0^\pi \frac{\cos(kx)}{\cosh(ax)}\ dx
$$
there $k\in
\mathbb{Z}, a\in \mathbb{R}.$
Of course that is Fourier coefficient for the function $f(x)=\frac{1}{\cosh(ax)}.$
...

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79 views

### Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals:
$$\int_0^t J_l(s) e^{-iA(t-s)}ds$$
$$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$
with $l>0$, $t>0$, $\Re(A)>0$, ...

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votes

**0**answers

117 views

### Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?

The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...

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**1**answer

276 views

### Laurent expansion of a principal value integral

Let $f(t)$ be a nice Hölder continuous function. Also, suppose that $f$ is even. I'm interested in evaluating integrals of the form:
$$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$
...

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**1**answer

117 views

### Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function.
Thanks!

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198 views

### A multiple integral

Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos ...

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**2**answers

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### Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:
Is it possible to express ...

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**4**answers

422 views

### Computing $\int_0^T e^{itA}Be^{-itA} dt$ without an infinite series

I'm hoping to compute the following integral: $\int_0^T e^{itA}Be^{-itA} dt$ where $iA, iB$ are traceless anti-Hermitian matrices (i.e. $\mathfrak{su}(n)$). I have found the following form for the ...

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229 views

### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...

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294 views

### Integral of sin(x)/sqrt(x) from 0 to \pi [closed]

How to calculate improper integral $\int_{0}^{\pi}{\frac{\sin{t}}{\sqrt{t}}dt}$?

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### Luria-Delbrueck model with deterministic gompertz growth of the wild type

i'm currently looking at a problem from population dynamics. The assumption is that a colony of wild-type cells growth according to the "gompertz-function"
$f(t)=m^{1-\exp(-\lambda_0 t)}$
where $m$ ...

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240 views

### Can it be decided whether $\int\root 3 \of{\cos^2(t)}\,dt$ is expressible by elementary functions? [closed]

I would like to decide by methods of Differential Algebra whether the integral $\int\root 3 \of{\cos^2(t)}\,dt$ contrary to the output of CAS Mathematica Online Integrator might
be expressible by ...

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**1**answer

301 views

### Conjectured closed form for definite integral

Let $K(x)$ be the complete elliptic integral of the first kind
(the argument is the parameter $m = k^2$).
Let $$ A = \int_0^1 \arcsin(K(x)) dx$$
With precision $1000$ decimal digits $\Re A = ...

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votes

**2**answers

232 views

### How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim ...

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**1**answer

128 views

### Question about the derivative of a fuctional

I have this lemma+proof and i dont understand why it follows from $J'(u_n)\rightarrow 0$ that $-\Delta_p u_n- f(x,u_n)\rightarrow 0$ such that
$J(u)=\frac1p\int_{\Omega} |\bigtriangledown u|^p ...

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25 views

### Convergence integral in probability

Let $X_1,\dots,X_n$ be i.i.d with distribution function $F$. Let $\hat F_n$ be their modified empirical distribution function, i.e.,
$$
\hat F_n(x)=\frac1{n+2}\left(1+\sum_{i=1}^n1_{\{X_i\le ...