Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.40681

Integration by differentiation

Alexandre M. Afonso
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note, we address a traditional pedagogical asymmetry between algorithmic differentiation and heuristic integration by introducing an extra parametric-order differential framework. By treating the order of differentiation as a parametric axis (discrete or continuous), classical integration is reframed as a local differential evaluation at the non-positive boundary level. This analytical shift alleviates some traditional routines of integration by parts, by replacing the quadrature of a product of elementary functions with a direct differential series that either truncates into a finite algebraic sum or resolves through cyclic symmetries. This parametric-order approach also bridges the gap with fractional calculus, showing that non-integer derivatives emerge as a seamless extension of the same operational dimension. By reducing global quadrature to local differentiation, this note offers an efficient, conceptual tool for undergraduate analysis.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexandre M. Afonso. 2026-10-26. Integration by differentiation. https://arxiv.org/abs/2610.40681

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

KEEP EXPLORING

Related papers

arXiv2026

The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $\left(p,q\right)$-Adic Fourier Analysis and Wiener's Tauberian Theorem

This paper gives an overview of $\left(p,q\right)$-adic Fourier theory - the Fourier theory of functions from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes - which we then use to prove a novel $\left(p,q\right)$-adic generalization of Norbert Wiener's celebrated Tauberian Theorem. Letting $K$ be a metrically complete, algebraically closed local field of residue characteristic $q$, letting $C\left(\mathbb{Z}_{p},K\right)$ be the Banach space of continuous functions $\mathbb{Z}_{p}\rightarrow K$, and letting $dμ$ be a $\left(p,q\right)$-adic measure (a continuous linear functional $C\left(\mathbb{Z}_{p},K\right)\rightarrow K)$, the $\left(p,q\right)$-adic Wiener Tauberian Theorem (WTT) we prove establishes that the density of the span of translates of $dμ$'s Fourier-Stieltjes Transform implies the non-vanishing of the Radon-Nikodym derivative of $dμ$ at all points in $\mathbb{Z}_{p}$ where the derivative exists in $K$.

math.GM↗
arXiv2026

Indefinite integrals of Bessel and Struve functions with half-integer indices, and incomplete gamma functions with integer indices, all multiplied by $Exp(-ax^{2})$ and divided by powers

Indefinite integrals are found for 63 half-integer Bessel and Struve functions, and incomplete gamma functions with integer indices, each multiplied by $Exp(-ax^{2})$ and divided by powers. A series solution is given for the individual terms (of any inverse power) in such functions, split into even and odd portions. Eight integrals are given involving these series. Three infinite sets of auxiliary integrals are also given.

math.GM↗