**2**

votes

**0**answers

45 views

### Wave trace on $\mathbb{S}^1$- $\langle w,\varphi\rangle=\int_{\mathbb{S}^ 1} (\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}) \varphi(t) dt$ [on hold]

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...

**4**

votes

**1**answer

60 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 ...

**1**

vote

**0**answers

43 views

### A sufficient condition of the following integral to be positive

Suppose $G=(A,B,E)$ is a bipartite graph whose partition has the parts $U=\{u_1,\cdots,u_m\}$ and $V=\{v_1,\cdots,v_n\}$. Consider the following integral
...

**2**

votes

**0**answers

151 views

### Euler-Maclaurin Formula Review

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
Generalizations to broader ...

**2**

votes

**0**answers

77 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 ...

**1**

vote

**0**answers

64 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 $f^{-1}\in C^{1}([a,b])$, ...

**1**

vote

**0**answers

21 views

### Expectation of two identical log-normal distributions [migrated]

I would like to compute the conditional expectation (on an interval from $c$ to $\infty$) of the minimum of two log normal distributions.
Denote $X_1$, $X_2 \sim LN(0, \sigma)$, the associated ...

**0**

votes

**0**answers

54 views

### Does $\int_x^y fdg=0$ for all $x<y$ and fixed positive $f$ imply $g$ is constant? [closed]

Assume that $f:[a,b]\to(0,1]$ is a given continuous and strictly decreasing function and $g:[a,b]\to\mathbb R$ is continuous and of bounded variation. Moreover we know that the Riemann-Stieltjes ...

**0**

votes

**0**answers

28 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

171 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 ...

**1**

vote

**2**answers

253 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

159 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

119 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

43 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

71 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

19 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

158 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

89 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

220 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

264 views

### Two similar integrals

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

**1**

vote

**0**answers

50 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

60 views

**4**

votes

**1**answer

130 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

612 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

60 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

73 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 ...

**8**

votes

**0**answers

133 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

128 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

135 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**

vote

**1**answer

70 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

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

**5**

votes

**1**answer

314 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

35 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

72 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

132 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

676 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

289 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

411 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

207 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

217 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

91 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

59 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

167 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

238 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

157 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

83 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

115 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

178 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 ...