The integration tag has no wiki summary.

**7**

votes

**2**answers

232 views

### Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...

**8**

votes

**2**answers

270 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
...

**1**

vote

**3**answers

2k views

### “Riemann–Stieltjes derivative”?

Can you define a "derivative" operator such that its antiderivative F(x) of f(x) can be used in the sense of F(b)-F(a) to calculate the Riemann–Stieltjes integral of f(x)?
Perhaps it would be related ...

**0**

votes

**0**answers

86 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 ...

**0**

votes

**0**answers

28 views

### numerical solution of volterra integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation
$$
\int_{a}^{x}f(s)g(s,x)ds=h(x)
$$
...

**5**

votes

**1**answer

263 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:
$$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$
...

**1**

vote

**2**answers

112 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**

votes

**1**answer

150 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 ...

**0**

votes

**0**answers

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

**2**

votes

**0**answers

77 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 ...

**0**

votes

**1**answer

91 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!

**8**

votes

**1**answer

237 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 ...

**-1**

votes

**0**answers

53 views

### Integrability of the logarithm wrt a finite Borel measure [migrated]

I have a finite Borel measure $d\phi$ on $(0,1)$, i.e. $\int_0^1 d\phi(x) < \infty$.
Is it also true that $\int_0^1 \log (x) d\phi(x) < \infty$?
The function $\log$ is integrable at 0, so ...

**3**

votes

**1**answer

264 views

### Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...

**10**

votes

**3**answers

2k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

**2**

votes

**0**answers

100 views

**43**

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 ...

**9**

votes

**0**answers

160 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 ...

**1**

vote

**0**answers

181 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 ...

**15**

votes

**4**answers

2k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

**6**

votes

**4**answers

390 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 ...

**69**

votes

**17**answers

14k views

### Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...

**2**

votes

**0**answers

235 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**

votes

**0**answers

16 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

220 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

**2**answers

163 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 ...

**2**

votes

**1**answer

270 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 = ...

**0**

votes

**1**answer

123 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

**0**answers

19 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 ...

**19**

votes

**2**answers

1k views

### What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?

I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...

**4**

votes

**2**answers

137 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 ...

**0**

votes

**1**answer

92 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

123 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 ...

**1**

vote

**1**answer

135 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 ...

**3**

votes

**0**answers

62 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 ...

**3**

votes

**1**answer

303 views

### Integral wrt probability measure

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...

**2**

votes

**1**answer

81 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 ...

**-2**

votes

**1**answer

102 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 ...

**3**

votes

**1**answer

95 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 ...

**2**

votes

**2**answers

326 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

427 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 ...

**2**

votes

**3**answers

352 views

### Defining the integral of a function using the product measure

Imagine that we're trying to define the expression
$$\int_U f(x)dx$$
in a rigorous way.
Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...

**12**

votes

**5**answers

1k views

### An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...

**5**

votes

**1**answer

123 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

112 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

182 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) &=& ...

**3**

votes

**1**answer

300 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 ...

**14**

votes

**2**answers

467 views

### Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here:
Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate
$$I_n
:=
...

**1**

vote

**1**answer

195 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 ...

**1**

vote

**2**answers

96 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 ...