**0**

votes

**0**answers

40 views

### Proving that a complex expression of integrals is increasing in a given parameter

Let $f$ and $F$ denote the respective pdf and cdf of a probability
distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real
$c$ such that $c\geq0$, and $\rho\geq0$.
We want to prove ...

**1**

vote

**1**answer

84 views

### Can this equality hold for a nonzero $b$?

Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...

**1**

vote

**1**answer

84 views

### Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$...

**-1**

votes

**0**answers

94 views

### Summing a series of integrals [closed]

EDIT: This IS related to my research (investigating representations of the harmonic mean)and I gave the wrong formula for the sum the first time around. The sum formula has been amended below.
I ...

**6**

votes

**2**answers

118 views

### Legendre Polynomial Integral

How can I evaluate
$$ \int_{-1}^1 P_n(x)P_l(x)x^k dx $$
when $k$ is even?
Or what might be a source where I could find integrals like this?

**10**

votes

**0**answers

216 views

### Integrability property of polynomials in several variables

This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...

**1**

vote

**1**answer

92 views

### How to prove this integral equality with limits

The following result is from a paper. The author says it is not hard to show that:
$$\lim_{t\to 1}\dfrac{1-t}{\sqrt{1+pt}}\int_{0}^{t}\dfrac{a(1+pa)}{(1-a)^2}\left(4a\left[1-\left(\dfrac{1-t}{1-a}\...

**1**

vote

**0**answers

227 views

### Prove this function is increasing

I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...

**1**

vote

**1**answer

103 views

### Convergence of Riemann integrals that do not hold for Lebesgue integrals

I am interested in convergence results that are true for Riemann-integrable functions but fail for Lebesgue-integrable functions. I know three of these, which happen to be closely related.
...

**0**

votes

**0**answers

20 views

### Splitting the region and estimating fractional Sobolev norms

x-post from math.stackexchange (http://math.stackexchange.com/q/1836766/349671), since the question arose from reading through a scientific paper:
I've been reading the paper "On the Bourgain, Brezis,...

**1**

vote

**0**answers

29 views

### Integral of Daubechies wavelets [closed]

For Daubechies wavelets according to this paper (above eq 19) this relation holds
$$
\int_{-\infty}^{-\infty} \phi(2x-i)\phi(2x-j)dx = \frac{1}{2} \int_{-\infty}^{-\infty} \phi(x-i)\phi(x-j)dx
$$
...

**6**

votes

**0**answers

127 views

### Symmetry of function defined by integral

(Originally posed in Math.SE in Jan 2013. Received no complete answers as of yet.)
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\...

**4**

votes

**1**answer

66 views

### Usable Change-of-Variables Formula for Hausdorff Measure

Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...

**2**

votes

**0**answers

50 views

### Do any of these integrals have closed forms in terms of special functions?

I've been looking at nonelementary integrals of the form $\frac{1}{f(x) + g(x)}$, where $f$ and $g$ are simple but different enough to be interesting. Mathematica can't evaluate any of these integrals,...

**4**

votes

**0**answers

208 views

### Verifying a source that lacks a citation

In this German Mathematics Wikibook page, formula $0.5$ lists the following equation
$$\int_0^1 \sin(\pi x) x^x (1-x)^{1-x} \ \mathrm{d}x = \frac{\pi e}{24}$$ as supposedly attributed to Ramanujan (...

**2**

votes

**0**answers

36 views

### Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...

**1**

vote

**0**answers

43 views

### Equivalent Definitions of the Gaussian Surface Measure for Regular Sets

I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....

**3**

votes

**0**answers

77 views

### interpretation of a singular integral

There is a post on MSE about a principal value integral in this paper. It has not received much attention even with a bounty, and since it concerns a published paper, I believe this is a better forum ...

**11**

votes

**1**answer

172 views

### Harmonic analysis, compute that this integral tends to $0$

We have the following setting.
$U$ is a bounded Lipschitz domain in the complex plane.
Consider the following classical Dirichlet problem for the Laplace operator:
$$\begin{align}
\Delta{}u&=0 \...

**1**

vote

**2**answers

245 views

### How can we obtain the $-\frac{4\pi}3\mu(x)$ term?

Given the expression
$$K_{ik} := \frac{\partial}{\partial x_k} \int_{\mathscr X} \frac{y_i-x_i}{|y-x|^3} \mu(y) dy,$$
where $\mathscr X=\mathbb R^3$, how does one derive the expression
\begin{align}
...

**4**

votes

**1**answer

67 views

### Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...

**2**

votes

**2**answers

382 views

### Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely
Specifically, I would like to ...

**3**

votes

**0**answers

80 views

### The integral of $\exp(-|x-a|)$ over an even dimensional sphere

I'm after a reference for an integral. For $m$ a positive integer and $R>0$ let $S^{2m}_R\subset \mathbb{R}^{2m+1}$ denote the radius $R$ sphere of dimension $2m$. Suppose that $a$ lies inside ...

**0**

votes

**0**answers

78 views

### Integration and Inverse Function Theorem

Apologies if this sounds too silly for advanced math people here. It's long since I moved from mathematics to medicine and this problem appears in my research.
For an $f^{-1}\in C^{1}([a,b])$, ...

**0**

votes

**0**answers

34 views

### Generalizing Integration by parts for general bounded continous measure

Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...

**2**

votes

**1**answer

187 views

### Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...

**1**

vote

**2**answers

301 views

### Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial.
For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$
we have the ...

**0**

votes

**1**answer

168 views

### Formula for an integration on $\mathbb{Q} \cap [0,1]$

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting:
$$ \...

**5**

votes

**1**answer

127 views

### Showing the positivity of a singular integral that came up in circle method

Let $F(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous form. Let
$$
I(\alpha) = \int_{[0,1]^n} e^{2 \pi i F(\mathbf{x}) \alpha} dx_1...dx_n.
$$
Then the singular integral is ...

**1**

vote

**0**answers

50 views

### First variation on double integral [closed]

Currently I am trying to fully understand the paper of munk1921.
In the derivation of the minimum induced drag theorem it is at one point stated (p.378) that in order to minimize drag the following ...

**1**

vote

**1**answer

80 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

21 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

167 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 $\Omega_i$'...

**1**

vote

**0**answers

94 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

247 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

270 views

### Two similar integrals

Let $n$ be a given even positive integer. We have the following integral
\begin{align}
\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...

**1**

vote

**0**answers

54 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

68 views

### A complicated integral inequality

How can we bound this integral:
$${\displaystyle \int_{-1}^{1}2\left[\dfrac{1}{4}-\dfrac{1}{4\left(1-\xi^{2}\right)}\left(1-\dfrac{\xi^{2}}{2}\right)^{2}\right]\left(\hat{f}\left(\xi\right)\right)^{2}...

**4**

votes

**1**answer

133 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

620 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 i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...

**0**

votes

**0**answers

64 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 \...

**3**

votes

**0**answers

81 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 ...

**13**

votes

**1**answer

289 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

141 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

138 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 $a \neq 0$...

**1**

vote

**1**answer

73 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

138 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 $x:K\to{\mathbb C}$, ${\...

**6**

votes

**1**answer

325 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 $(V,\mathcal{V})$...

**1**

vote

**0**answers

37 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

94 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 ...