The integration tag has no wiki summary.

**-3**

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34 views

### Find the integral [on hold]

How can we find the integral of the 1/(1+x^4) in the interval -infinity to +infinity.I tried to find and got it to be pi/sqrt(2). Am I correct? Please help me with an appropriate method. I tried to ...

**1**

vote

**1**answer

58 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 ...

**0**

votes

**0**answers

51 views

### How to prove $\lim _{ \delta\rightarrow {0}^{+}}\int_{a}^{b} F_{\delta}(x)dx=0 ? $ [on hold]

This question comes from http://math.stackexchange.com/questions/982231/a-function-fx-that-riemann-integrable-on-a-b
Define a function $f(x)$ that Riemann integrable on $[a,b]$.
Let
...

**0**

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**0**answers

47 views

### Guess A Property Of The Integral Average Value Function [on hold]

Let $f$ be a function that is defined on $[a,b]$ and Riemann integrable on $[a,b]$.
Def 1.
$$\hat f(x)=\begin{cases}
f(x),& \text{if }x\in[a,b], \\
f(a),& \text{if }x<a, \\
...

**0**

votes

**0**answers

141 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[ ...

**2**

votes

**0**answers

70 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 $X$ ...

**0**

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**0**answers

92 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
$$
...

**0**

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**0**answers

59 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
...

**5**

votes

**1**answer

89 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 ...

**0**

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**0**answers

75 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 ...

**2**

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**0**answers

28 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
...

**1**

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**0**answers

38 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 ...

**0**

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**0**answers

33 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 ...

**6**

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**0**answers

51 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 ...

**7**

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**3**answers

260 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 ...

**6**

votes

**2**answers

146 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 ...

**0**

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**0**answers

50 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 ...

**0**

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**0**answers

215 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**

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**0**answers

52 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

93 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**

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**0**answers

28 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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**0**answers

30 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, ...

**3**

votes

**0**answers

71 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 ...

**1**

vote

**1**answer

113 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 ...

**3**

votes

**1**answer

193 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**

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**0**answers

92 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 ...

**1**

vote

**2**answers

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

**0**

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**0**answers

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

**3**

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**0**answers

107 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

272 views

### Laurent expansion of a principal value integral

Let $f(t)$ be a nice HÃ¶lder 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

109 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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110 views

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186 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 ...

**49**

votes

**2**answers

2k 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

419 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

227 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

261 views

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

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

**0**

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**0**answers

17 views

### 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 $m$ ...

**2**

votes

**0**answers

238 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

291 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

198 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 ...

**0**

votes

**1**answer

127 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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**0**answers

22 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 ...

**0**

votes

**1**answer

101 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 ...

**2**

votes

**1**answer

134 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

240 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

66 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

85 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

156 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 ...

**-2**

votes

**1**answer

103 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 ...