**0**

votes

**0**answers

219 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 ...

**1**

vote

**0**answers

138 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?
...

**0**

votes

**1**answer

112 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) := ...

**0**

votes

**0**answers

53 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 ...

**0**

votes

**0**answers

36 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, ...

**1**

vote

**1**answer

332 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 ...

**4**

votes

**1**answer

248 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 $p$, $q$ ...

**0**

votes

**0**answers

109 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 ...

**2**

votes

**2**answers

164 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 $f(x)=\frac{1}{\cosh(ax)}.$
...

**1**

vote

**0**answers

113 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 $l>0$, $t>0$, $\Re(A)>0$, ...

**4**

votes

**0**answers

172 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 ...

**5**

votes

**1**answer

302 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:
$$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$
...

**0**

votes

**1**answer

201 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!

**2**

votes

**0**answers

129 views

**11**

votes

**0**answers

310 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 ...

**62**

votes

**2**answers

4k views

### 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 ...

**8**

votes

**4**answers

535 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 ...

**1**

vote

**0**answers

241 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 ...

**2**

votes

**0**answers

840 views

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

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

**2**

votes

**0**answers

261 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 ...

**2**

votes

**1**answer

414 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 = ...

**2**

votes

**2**answers

268 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 ...

**-1**

votes

**1**answer

144 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 ...

**0**

votes

**1**answer

127 views

### Monte Carlo estimator with autocorrelated samples

Given an integration problem $I=\int{f(x)dx}$, we can construct an ordinary Monte Carlo estimator as
$E[I]=\sum\limits_i\frac{f(x_i)}{p(x_i)}$
where the samples $x_i$ are usually i.i.d. and drawn ...

**4**

votes

**3**answers

232 views

### Quadrature formula max accuracy

I'm looking for a maximum accuracy quadrature formula:
$$
\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)
$$
I don't know exactly if it's Trapezoidal rule which has the ...

**2**

votes

**1**answer

266 views

### To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form
$$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$
where $\mu(\alpha)$ is a non decreasing function ...

**8**

votes

**1**answer

284 views

### When is this multiple integral finite?

Consider the following integral:
$$
I_k(\alpha)=\int_{[0,1]^k}|x_1-x_2|^{\alpha}|x_2-x_3|^{\alpha}\ldots|x_{k-1}-x_k|^{\alpha}|x_k-x_1|^{\alpha}d\mathbf{x}.
$$
where $k=2,3,4,\ldots$
The question is ...

**3**

votes

**0**answers

104 views

### What are the criteria for an elementary function to be infinitely integrable in elementary functions?

What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function?
Is there a way to check whether a given elementary ...

**2**

votes

**1**answer

169 views

### Is an integral against a probability measure in the convex hull of the range?

This may be really obvious but I don't see it. Let $f:\Omega \to \mathbb R^n$ be integrable with respect to a probability measure $\mu$. Does it follow that $\int_\Omega f \, d\mu$ is in the convex ...

**4**

votes

**2**answers

348 views

### Integrals of two Bessel functions of the first kind and a modified bessel function of the second kind

I'm searching for a suitable (hopefully simple enough) solution to the following form of integral:
$$\int_0^\infty \mathrm{d}x~x^n J_\nu(a x) J_\nu(b x) K_\mu(c x) $$
Where $n$, $\nu$, and $\mu$ are ...

**-3**

votes

**1**answer

105 views

### Suppose I know $\int h(t) dt = H(t)$, is there a way to find $\int h(t)^N dt$?

I am trying to find the -1 moments of sum of N geometric random variable, i.e. $E[\frac{1}{\sum_{i=1}^N X_i}]$
Suppose the probability mass function is $f_X(x) = (1 - p)^{x - 1} p$
The moment ...

**14**

votes

**2**answers

449 views

### On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation
...

**0**

votes

**1**answer

159 views

### Is any derivative of $f_1^x f_0^{1-x}$ w.r.t. $x$ integrable?

For $f_0$ and $f_1$ two continuos probability density functions on $\mathbb{R}$, by Hölder, I know that $f_1^x f_0^{1-x}$ is integrable on $\mathbb{R}$, where $0 \leq x \leq 1$. Let $l=f_1/f_0$, then ...

**3**

votes

**1**answer

241 views

### Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...

**1**

vote

**1**answer

141 views

### Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)?
That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain ...

**3**

votes

**2**answers

404 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

**4**

votes

**2**answers

511 views

### I don't understand behavior of this integral, help!

In an answer to a question I needed the following integral:
$$
f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt;
$$
it represented deviation from modularity of some other function. However I noticed ...

**1**

vote

**1**answer

380 views

### Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum
$$\sum_{i=1}^n f(x_i) w_i$$
where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...

**5**

votes

**1**answer

141 views

### Approximations of the identity on Lie groups and homogenous spaces

I'm looking for a nice (and preferably classic or book) reference for the following type of result:
Consider a transitive action of a compact Lie group $G$ on a compact manifold $M$ and a continuous ...

**1**

vote

**0**answers

124 views

### Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...

**3**

votes

**0**answers

250 views

### How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& ...

**1**

vote

**2**answers

152 views

### Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...

**1**

vote

**1**answer

222 views

### Request for help with two integrals

It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...

**3**

votes

**2**answers

691 views

### Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...

**3**

votes

**1**answer

320 views

### Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]

Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n.
We know that $f\equiv 0$. It's call Hausdorff theorem.
This theorem is wrong on $\mathbb{R^+}$, a ...

**4**

votes

**0**answers

148 views

### The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.
As ...

**4**

votes

**2**answers

149 views

### Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?

The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th ...

**4**

votes

**0**answers

261 views

### Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by
\begin{equation}
...

**2**

votes

**0**answers

110 views

### How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper.
Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$
On page 5 of ...

**1**

vote

**1**answer

101 views

### What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer.
Integral 1.8.2.4:
$$
\int_0^x x^{\nu+1} ...