Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1007.3460

Generalization of the Glasser - Manna - Oloa integral and some new integrals of similar type

Niyaz M. Vildanov
View original source ↗Read source document ↗Export metadata ↓

Abstract

As was shown in the previous works by other authors, Glasser - Manna - Oloa integral arise in the study of the Laplace transform of the dilogarithm function and can be evaluated in a closed form. In this article, we give a one parametric generalization of the Glasser - Manna - Oloa integral. The method employed in the course of derivation allows to obtain some new integrals of similar type. These include also integral representation of the Hurwitz zeta function and some beautiful formulae involving the logarithm of the gamma function of the argument -ix+ln(2cosx).

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Niyaz M. Vildanov. 2010-07-20. Generalization of the Glasser - Manna - Oloa integral and some new integrals of similar type. https://arxiv.org/abs/1007.3460

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Weighted inequalities in ergodic theory via transference

We first extend Calderón's transfer principle to weighted spaces in various different settings under suitable assumptions. Then we apply our results for some inequalities on the real line obtained by the author to prove corresponding inequalities in ergodic theory and ergodic $H^1$ spaces as well.

math.CA↗
arXiv2026

Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains

Given a uniform domain $Ω\subset {\mathbb R}^d$, we resolve each element of a suitably defined class of Calderòn-Zygmund (CZ) singular integrals on $Ω$ as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journé theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case $Ω={\mathbb R}^d$ with Lebesgue measure. Our characterization covers the case of compressions to $Ω$ of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space $W^{1,p}(Ω)$, $p>2$.

math.CA↗