A simple method of computing Riemann-type integrals
Abstract
The "Riemann-type integrals" of the title are not a single new integral, but the family of classical constructions -- Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô -- unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on ℝ; it vanishes in the limit under a quadratic-variation condition Q(π)→0 on ℂ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction -- except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point -- that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts -- as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on ℝ and ℂ; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.
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BibTeX
@misc{tubaro2026simple,
title = {{A simple method of computing Riemann-type integrals}},
author = {Luciano Tubaro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05562},
eprint = {2609.05562},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05562},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A simple method of computing Riemann-type integrals
AU - Luciano Tubaro
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.05562
UR - https://arxiv.org/abs/2609.05562
AB - The "Riemann-type integrals" of the title are not a single new integral, but the family of classical constructions -- Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô -- unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on ℝ; it vanishes in the limit under a quadratic-variation condition Q(π)→0 on ℂ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction -- except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point -- that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts -- as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on ℝ and ℂ; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.
ER -