# Tagged Questions

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### Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
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### exponential integral $e^{x/t^{3}}$ and floor function

Is it possible to give a closed form for the integral $$\int_{0}^{\infty} \frac{dt}{t^{3}}\rho (t)e^{-\frac{x}{n^{2}t^{2}}}$$ where $\rho (t) = t- \left\lfloor t\right\rfloor$ is the fractional ...
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### Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
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### Lebesgue integrability and Kurzweil Henstock integrability

Why it's possible to integrate the function: $$f(x)=\frac{1}{x}\sin\left(\frac{1}{x^\alpha}\right)$$ using Kurzweil Henstok integral while it's not Lebesgue integrable because the singularity in ...
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### “Values” of divergent integrals

Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
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### Convergence of the integral of step functions

This is a question about the proof of Lemma A in §16 of the book Functional Analysis by F. Riesz and B. Sz.-Nagy. Lemma A: For every sequence of step functions $\{\varphi_n\}$ which decreases to ...
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### Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral

Related to this and this questions. Basically got definite integral that experimentally equals $\zeta(s)$ both numerically and symbolically. Closed form for the indefinite integral is known, but I ...
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### Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...
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### Coutour Integral of Gamma Functions

How do I solve the Integral $$\frac{1}{2\pi j} \oint \frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{ (2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$ This integral is an inverse ...
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### Stokes theorem for manifolds without orientation?

Hello! In textbooks Stokes theorem is usually formulated for orientable manifolds (At least I couldn't find any version not using orientability). Is Stokes theorem: ...
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### Why is there a formula for symbolic differentiation (chain and product rules) but not for symbolic integration? [duplicate]

Possible Duplicate: Why is differentiating mechanics and integration art? There is a formula for the derivative of any product, composite or sum of functions, in terms of the derivatives of ...
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### Nontrivial trivial integrals

I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that. Consider two propositions in geometry: ...
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### Sum involving binomial coefficients

I have the following sum $\sum_{j=1}^K {K \choose j} (-1)^{j+1}/j$. Now I can write this as the integral $\int_{-1}^0 \frac{(1+x)^K - 1}{x} dx$. However, I wonder whether there is a closed form ...
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### convergence of sets and limit of an integral

Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets. Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function. Let $s:Y\rightarrow X$ be a function (not necessarily continuous). ...
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### An infimum of integrals of a positive function.

Hi, I have a question concerning integration theory I can't figure out, maybe someone can help: Fix $\varepsilon>0$ and consider $\delta \colon [0,1] \to (0,\infty)$ measurable. Is it then true ...
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### definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011). There, they work on a Hilbert space $H$ and on the ...
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### What are integration on fractal? [closed]

Who can explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK ...
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### Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution. When I measure the convergence rate ...