# 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, ...

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### 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')\...
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 ...
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 ...
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}^{\...
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
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 ...
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 $f'$...
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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 ...
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 ...
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 ...
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 ...
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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 $C=e^\gamma$ ...
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} ...
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
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{...
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....
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 ...
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 ...
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). ...
38 views

1 vote