**1**

vote

**1**answer

64 views

### $p$-adic Dirac measure as a weak limit

The standard Dirac delta is a generalised function (or measure, or distribution...) $\delta(x)$ which can be seen as a weak limit functions $\delta_n(x)$ spiked at the the origin, in the sense that:
...

**1**

vote

**0**answers

18 views

### Bivariate integration with the range of one variable shrinking to a point

Let $f(x,y)$ be a measurable function defined on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. Define $C_{\epsilon} = \{y:d(y, y_0)\leq \epsilon\}$, then can we say for sure that the integration
$$
...

**2**

votes

**0**answers

134 views

### A question about multidimensional integral

Consider the function
$$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$
Is there a necessary condition on the ...

**1**

vote

**0**answers

193 views

### Existence of probability distributions/measures/spaces and mathematical expectations in some functional spaces [closed]

Please see my questions:
What's the probability distribution of a deterministic signal? (functional integrals in probability theory)
Is there a Bayesian theory of deterministic signal? Prequel ...

**1**

vote

**0**answers

85 views

### Integration of Bessel Function of the first kind

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows:
$$\int_{0}^\infty F(x)[x^3*B*J_0(xy)+x^4*J_1(xy)]dx=G(y)$$
where $B$ is a ...

**2**

votes

**0**answers

189 views

### Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question:
What's the probability distribution of a deterministic signal? (functional integrals in probability theory)
Clearly my question looks at the same time fairly ...

**4**

votes

**1**answer

256 views

### Two similar integrals

Let $n$ be a given even positive integer. We have the following integral
\begin{align}
...

**1**

vote

**0**answers

47 views

### Complex integration over 1-singular chain

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...

**1**

vote

**0**answers

54 views

**4**

votes

**1**answer

126 views

### When this Ad-invariant function on a Lie algebra is zero?

Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|_{\frak g}$ so that $\frak g$ ...

**5**

votes

**1**answer

601 views

### Is the following integral nonzero?

Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq ...

**0**

votes

**0**answers

52 views

### Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let
\begin{equation}
I = \int_D g(\textbf{x})d\textbf{x},
\end{equation}
where $D \subset ...

**-1**

votes

**0**answers

34 views

### How to prove that arc segment vanishes

I have this integral:
$$iNx^a\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{Ne^{i\theta}}e^{i\theta}}{\Gamma(Ne^{i\theta}+a-m)\zeta(2Ne^{i\theta}+2a+n)\sin (\pi \left(a+Ne^{i\theta}\right))} ...

**3**

votes

**0**answers

66 views

### Divergence theorem on stratified spaces

It is very common in physics and engineering to apply the divergence theorem
to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating ...

**3**

votes

**0**answers

265 views

### Wrongful conviction Bayesian argument in need of integral-solving talent [migrated]

Motivation/background: The following integrals were part of an argument in my PhD thesis that a certain piece of forensic evidence has no inculpatory value. I gave up trying to solve them analytically ...

**8**

votes

**0**answers

117 views

### Summation of series involving $\sinh$ of a square root

Consider the following series:
$$
S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})}
$$
From the physical ...

**5**

votes

**2**answers

123 views

### Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...

**5**

votes

**1**answer

132 views

### Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq ...

**-1**

votes

**0**answers

34 views

### Metrics for a Wireless Sensor Network coherent model

If we describe the signal of a Wireless Sensor Network through the following coherent Gaussian model
$$ S(x,y; \boldsymbol{r}_i) = \sum_{i=1}^N \exp\left[-\frac{(x-x_i)^2+(y-y_i)^2}{2 ...

**1**

vote

**1**answer

69 views

### Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers):
$$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$
I could not find it in ...

**3**

votes

**1**answer

124 views

### A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$

I asked this at math.stackexchange, but nobody answered.
Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ...

**1**

vote

**0**answers

69 views

### Finding a closed form for a certain double integral

I am working with bivariate and accurate Birnbaum-Saunders distribution to find the probability density function of a particular model for this, I would like to find a closed form for the full below:
...

**5**

votes

**1**answer

309 views

### Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and ...

**1**

vote

**0**answers

33 views

### Can there be a nonzero period integral of this form?

I have been trying to compute the following integral:
$$\displaystyle \int_{\gamma} \frac{g \Omega_{\mathbb{P}^3}}{f^2}$$
where:
$\gamma$ is the torus cycle $\{ |z_1| =|z_2| =|z_3| =|z_4| =1 \}$,
...

**3**

votes

**0**answers

62 views

### Using and understanding the Atiyah-Bott localization theorem/integration formula

I posted this on r/math, but was told I might have better success here given the level of the question.
Basically, I need to learn how to use the localization theorem to compute integrals on ...

**4**

votes

**0**answers

128 views

### Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the ...

**2**

votes

**0**answers

565 views

### What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required.
In random signal theory, this distribution is typically a ...

**11**

votes

**2**answers

277 views

### How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?

For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...

**12**

votes

**3**answers

393 views

### Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define
$$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$
with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $.
Is it ...

**10**

votes

**1**answer

204 views

### Calculation of the integral related to the gravitational shock wave

The following integral
$$\int\limits_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$
can be found in the paper
Tevian Dray and Gerard 't Hooft, The ...

**6**

votes

**0**answers

214 views

### An inequality which involves a sum of integrals

Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad ...

**1**

vote

**1**answer

88 views

### A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...

**0**

votes

**0**answers

55 views

### Expansion of integral with Gaussian kernel

This question may be simple for you, so you are invited to address me to where I can read about it. I have an integral of the form
$$ \int f(\bf x, \bf x') g(\bf x') $$
where $$ f(\bf x , \bf x') = ...

**2**

votes

**0**answers

164 views

### How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral
...

**3**

votes

**1**answer

230 views

### Integral involving the gamma function

If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating
$$ \int_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter ...

**-2**

votes

**1**answer

98 views

### Is this intergral inequality valid? [closed]

Does the inequality $\int_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?

**3**

votes

**1**answer

94 views

### The volume of a region arising from planar linkages

Let $x_0,\dots,x_n$ be a collection of variable points in $\mathbb{R}^2$ and let $c>0$ be a fixed constant. Is there any way I could compute an upper bound of the volume of the region in ...

**1**

vote

**1**answer

155 views

### Dense subspaces of $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert ...

**1**

vote

**0**answers

78 views

### Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable?
This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion ...

**5**

votes

**0**answers

114 views

### Integral-like concepts

I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect.
The background is that I have proven a ...

**3**

votes

**1**answer

177 views

### An interesting integral to determine the sign

I would like to know Whether the integration
$\int_0^\infty\frac{s^{N_1+N_2}(2s^{N_1+1}-1)}{(1+s^{N_1+1})^4(1+s^{N_2+1})^2}ds$
is positive or negative? where $N_1,N_2$ are positive integers.
I ...

**1**

vote

**0**answers

78 views

### Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ ...

**3**

votes

**1**answer

90 views

### Characterization of $L^1(\text{SL}(3,\mathbb{R}))$ [closed]

Is there a characterisation of the integrable functions on SL($3,\mathbb{R}$) or an explicit expression for the Haar measure?

**4**

votes

**1**answer

121 views

### Mean value of a function associated with continued fractions

Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?

**0**

votes

**1**answer

164 views

### $\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...

**6**

votes

**1**answer

273 views

### Law of unconsious statistician: application in characteristic function

Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...

**21**

votes

**3**answers

514 views

### Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:
$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$
...

**1**

vote

**0**answers

47 views

### Integration involving modified bessel function, exponential and power

I need to find the following integration.
$$
\int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N
$$
where
$$
a>0, \quad N \geq 1
$$
Any help will be much appreciated.
BR
Frank

**0**

votes

**0**answers

75 views

### Integral involving modified bessel function of second kind, exponential and power

I need to compute the following integral.
$$
\int_0^a e^{-bx}\sqrt{4(a-x)}K_1(\sqrt{4(a-x))}dx\,.
$$
where $$ a>0$$
and $b$ can be greater than zero or less than zero but it is not a complex ...

**0**

votes

**0**answers

54 views

### Optimal growth of an oscillating integral

Let $f\in H^1(\mathbb{R}^3)$, with $f\equiv0$ inside a ball around the origin. For $t>0$, consider the following integral
$$I(t):=\int_{\mathbb{R}^3}e^{i|x|^2/t}\frac{f(x)}{|x|}dx$$
It`s easy to ...