# Questions tagged [integration]

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

1,392
questions

4
votes

1
answer

159
views

### When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...

3
votes

1
answer

128
views

### If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?

Let $a,b\in\mathbb R$ with $a<b$ and $f:[a,b]\to\mathbb R$. Assume that there exists a Riemann integrable function $g:[a,b]\to\mathbb R$ such that $f=g$ almost everywhere.
Then we can NOT conclude ...

9
votes

1
answer

479
views

### Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...

0
votes

0
answers

85
views

### An integral and a Jensen-type formula

For $f$ be analytic on the disc $\overline{D}(0,R)$ centred at $0$ with radius $R>0$ and such that $f(0)\neq 0$, then the following formula is well-known
\begin{align}
\frac{1}{2\pi}\int_{-\pi}^{\...

-5
votes

0
answers

92
views

### How can I calculate this integral: $ \int \frac {x+n}{\sin x+e^x} dx$? [closed]

I need to solve this integral:
$$ \int (x+n)/(\sin(x)+e^x) dx$$
I tried everything, from u-substitution to Feynman technique and it didn't work. Please Help!

1
vote

1
answer

155
views

### Average distance between points of lower dimensional simplices in $\mathbb R^n$

Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...

5
votes

1
answer

324
views

### On the Riemannian integrability of the bounded derivative

Let $f:[a,b]\to\mathbb R$ be a differentiable function with $f'$ bounded. According to this post, $f'$ is not necessarily Riemann integrable on $[a,b]$, see also Volterra's function.
I wonder, if $f'$...

2
votes

0
answers

91
views

### A generalised Young integral

Let $f: [0, 1] \to \mathbb R$ be a continuous function. The pointwise Holder exponent $H_f (x)$ of $f$ at $x \in [0, 1]$ is defined to be
$$H_f (x) := \sup \left \{ \alpha \in [0, 1] \, \big |\, \...

5
votes

0
answers

477
views

### Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?

Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacobian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill,
The integrals
$$\operatorname{sn}^2u,\operatorname{...

-1
votes

0
answers

146
views

### Interchange of max and integral

Consider a function $f: \mathbb{R} \times \Delta(\mathcal{Y}) \to \mathbb{R}$ and the following maximization problem:
$$\max_{Q \in \Delta(\mathcal{Y})} \int f(x,Q^{(P)}) \, dP(x),$$
where $P$ is a ...

0
votes

1
answer

154
views

### Can we integrate arbitrary rational functions of Jacobian elliptic functions?

We can integrate arbitrary rational functions of the trigonometric functions because of the tangent half-angle substitution (https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution). This led ...

-1
votes

0
answers

54
views

### Convolution of two modified Bessel functions

Does a closed formula (or power series expansion) for the following convolution exist?
$$I_{\nu}(x)=\int_{0}^{\infty} K_{\nu}(x-\tau)K_{\nu}(\tau)d\tau$$
Here $K$ stand for the modified Bessel ...

0
votes

0
answers

90
views

### Integration by parts over $R^n$ [migrated]

Let $\mu$ be a probability measure over $\mathbb{R}^n$. Let $f$ and $g$ be two real-valued functions on $\mathbb{R}^n$. I would like to compute
$$\int_{\mathbb{R}^n} \nabla f(x) g(x) \, d\mu(x). $$
I ...

2
votes

2
answers

177
views

### Limit of a integral whose integrand diverges under the limit

I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...

2
votes

1
answer

87
views

### Equivalent characterization of weak derivative in Bochner space

Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...

0
votes

0
answers

118
views

### Calculation of an integral from BCS superconducting theory

How do I calculate the following integral?
$$\int_0^{\infty} d z ~z^{1 / 2}\left[\frac{\tanh ((z-1) / 2 t)}{2(z-1)}-\frac{1}{2 z}\right]=\ln \left(\frac{8 C}{\pi e^2 t}\right),$$
where $C=e^\gamma$ ...

0
votes

0
answers

170
views

### Why is the sign of the integration negative?

Let
\begin{aligned}
I=\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx,
\end{aligned}
...

0
votes

1
answer

131
views

### Action of the Haar measure on the Heisenberg group

The Heisenberg group is $\mathbb{H}^N=\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the group operation
\begin{equation}
(...

1
vote

1
answer

117
views

### estimate involving Gaussian data

Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...

13
votes

1
answer

655
views

### How many integrals can give multiples of $\pi$?

This question notes a few families of rational functions whose integrals (from $0$ to $1$) give rational multiples of $\pi$. A fairly straightforward explanation is given there and in the related Math....

0
votes

0
answers

59
views

### Minimal polynomial of $\operatorname{cd}\frac{4K}{n}$

Define
$$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$
and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by
$$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$
on $z\in [0,2K]$ and by
...

0
votes

0
answers

49
views

### Characterizing functions that are limits of integrable lower-bounded functions

Let $X$ be a separable Hausdorff topological space, endowed with a positive finite Borel regular measure. Consider those (trivially measurable) functions $f : X \to \mathbb R$ such that their ...

3
votes

1
answer

147
views

### Closed-form expression for definite integrals involving modified Bessel functions K_1 and K_0

I am attempting to derive a closed-form expression for the following two integrals involving the modified Bessel functions $K_1$ and $K_0$, but I can't find a solution (I don't know if there is one). ...

0
votes

0
answers

38
views

### Lower bound for the fractional Sobolev norm of the Hermites function

For $r \in (0, 2)$, I am interested in a lower bound for the quantity :
$$I_r(n) := \int_{\mathbb{R}} |f_n(x)|^2 |x|^{r} dx$$
where $f_n(x) = (-1)^n (\sqrt{\pi} n! 2^n)^{-1/2} e^{x^2/2} \dfrac{d^n}{dx^...

-4
votes

1
answer

102
views

### An integral similar to the Delta function [closed]

I have an integral on the form
$\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau$
that I would like to simplify (or basically solve). This indeed comes from a problem ...

0
votes

0
answers

48
views

### How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?

We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...

2
votes

0
answers

127
views

### an upper bound for $L^1$ norm of the mollifier function

The standard mollifier function is defined as follows
$$f(x)=\begin{cases} 0 & \text{if } |x| \ge 1\\ \exp \left(-\cfrac{1}{1-x^2}\right) & \text{if } |x|<1.\end{cases}$$
It is well known ...

-1
votes

1
answer

106
views

### Riesz energy for open sets in dimension $1$

This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by
$$I_s(B(...

20
votes

1
answer

2k
views

### A difficult integral for the Chern number

Cross post from Maths stack exchange
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...

2
votes

1
answer

98
views

### Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave

I am struggling with the following problem. Let $f$ be a real smooth function:
strictly convex on $(-\infty,0)$,
strictly concave on $(0,\infty)$,
strictly increasing.
For $\sigma>0$, how can one ...

3
votes

1
answer

119
views

### Asymptotic behavior of the integral that contains $\delta$ function

The integral I want to calculate is defined as
$$
P(s)=\int_{-\infty}^{\infty}{\rm d}x\int_{-\infty}^{\infty}{\rm d}y\ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s\right)e^{-\left(x^2+y^2\...

3
votes

1
answer

87
views

### Surface integration w.r.t Hausdorff measure

I am reading at Evans' book Measure Theory and Fine Properties of Functions, Revised Edition, p. 165 and I can't see how one gets the transition from the first dotted (🔴) integral to the second one --...

4
votes

1
answer

185
views

### Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$

Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...

2
votes

0
answers

55
views

### Evaluation of a certain area

I asked a version of this question on Math Stack Exchange 6 days ago, but without any responses: The area of a certain region
I am interested in evaluating the area of the region defined by
$$A_{L_1, ...

0
votes

0
answers

20
views

### How to find condition on a linear operator $b: g \rightarrow g$ so that $b$ induces a twisted homomorphism on its simply connected Lie group

Let $G$ be a simply connected Lie group with Lie algebra $g$. Let $b: g \rightarrow g$ is a linear operator. Suppose I induce a map $B: G \rightarrow G$ from $b$ in following manner:
Let $e : g \...

1
vote

1
answer

117
views

### Convolution of a Hermite polynomial with Gaussian kernel

Let $H_n$ be the $n$th probabilistic Hermite polynomial of degree n and $\eta = \exp(-x^2/2)/\sqrt(2 \pi)$ be the standard Gaussian density.
I would like to compute the integral $f_n(x) = \int H_n(x - ...

6
votes

0
answers

102
views

### Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$

I am interested in showing that the integral
\begin{align}
& \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt]
= {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...

2
votes

1
answer

206
views

### Distance between root of $f$ and its Gaussian convolution

Let $f$ be a :
$f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$,
for all $x> 0,~f(x)>0$,
for all $x< 0,~f(x)<0$,
I am struggling to find a bound for the distance between the root of $f$ ...

2
votes

0
answers

67
views

### Computing a complex integral with many poles

For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i....

3
votes

2
answers

228
views

### Calculation of Riesz energy for balls

I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...

0
votes

0
answers

46
views

### A Riemann-Liouville differintegral for all entire Dirichlet L-series. Could it be simplified further?

It appears that the well-known relation between entire Dirichlet L-series and the Hurwitz zeta function $\zeta(s,a)$, with $k$ = modulus, $j$ = index of the Dirichlet character $\chi$:
$$(s-1)\,L\left(...

6
votes

2
answers

434
views

### Integral of the $\delta$ function

I want to calculate the integral defined as
$$
P(s)=\iint \mathrm dx \, \mathrm dy\ \ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s \right).
$$
The integration is taken within the rectangle $-...

3
votes

0
answers

217
views

### Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?

The well-known integral expression for the entire function:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...

3
votes

0
answers

154
views

### Do the difficulties in generalising Henstock-Kurzweil still exist if every subset of $\mathbb R^n$ is Lebesgue measurable?

There are apparently some difficulties generalising the Henstock-Kurzweil integral from functions of signature $\mathbb R\to\mathbb R$ to functions of signature $\mathbb R^n \to \mathbb R$. One ...

8
votes

1
answer

647
views

### A curious family of integrals that give $\pi$

I have noticed experimentally that:
$$\int_0^1 \frac{\color{red}{x}}{x^4+2x^3+2x^2-2x+1} \,dx=\color{blue}{\frac{\pi}{8}},\tag{1}$$
$$\int_0^1 \frac{\color{red}{1-x^2}}{x^4+2x^3+2x^2-2x+1} \, dx=\...

3
votes

1
answer

121
views

### The function $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m\,dy$ attains its maximum in $x=0.$

In this paper, Lemma 6, Pinsky proves that $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$ attains its maximum in $x=0.$ I didn't understand the proof very well, but ...

6
votes

0
answers

233
views

### Is there a uniform version of Lebesgue's differentiation theorem?

Let $\mu$ be a finite measure on $\mathbb R$ and $f,g : \mathbb R \to \mathbb R_{\geq 0}$ two measurable maps such that $\int_{x\in\mathbb R} f(x)\ \mu(dx) \leq 1$ and that $g(x) \leq 1$ for all $x$. ...

4
votes

1
answer

401
views

### Iterated Duhamel's formula for solutions of Boltzmann equation

My question comes from a computation in the paper Central limit theorem for Maxwellian molecules and truncation of Wild expansion. Specially, consider the following Boltzmann equation
$$\frac{\partial ...

2
votes

1
answer

156
views

### Is this integral solvable analytically?

I have this integral that comes from my research with some Fourier Transforms of spectrum functions:
$$ G(\tau) = \int_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c_1 - c_2 e^{-c_3 x} ) } dx $$
where $...

1
vote

0
answers

58
views

### Biot-Savart-like integral for a toroidal helix

The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...