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 It\^o -- 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 It\^o integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical It\^o 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 $\mathbb{R}$; it vanishes in the limit under a quadratic-variation condition $Q(\pi)\to0$ on $\mathbb{C}$ and for H\"older paths; and for a continuous semimartingale it survives as the quadratic variation producing the It\^o 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 It\^o 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 $\mathbb{R}$ and $\mathbb{C}$; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and H\"older (Young-Kondurar) cases; Part III extends it to the stochastic setting.