Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0912.3844

Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity

Richard J. Mathar
View original source ↗Read source document ↗Export metadata ↓

Abstract

Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over the first halve wave with series acceleration techniques for the alternating series co-phased to each of its points. The integrand is of the logarithmic kind; its branch cut limits the performance of integration techniques that rely on smooth higher order derivatives.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Richard J. Mathar. 2010-08-05. Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity. https://arxiv.org/abs/0912.3844

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↗