An Integration Algebra for Smooth Functions: Telescoping Formulas, Portal Structures, and the Parity Criterion
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866901889805713408 |
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| author | Wohnsiedler, Simon |
| author_facet | Wohnsiedler, Simon |
| contents | <p>v4 (22 May 2026): Minor cleanup.<br>(1) HAL identifiers removed from companion paper bibliography entry<br> — the HAL deposits were rejected due to affiliation policy and<br> the identifiers are no longer valid. Zenodo DOI is the sole<br> public identifier.<br>(2) Gel'fand–Dikii (1975) and Mikhailov–Novikov–Wang (2007) added<br> to bibliography. Originality Remark updated to acknowledge the<br> classical background of the parity criterion and precisely<br> enumerate new contributions.</p> <p>────────────────────────────────────────</p> <p>We develop a constructive theory of closed-form integration for smooth functions, in two layers. Layer I (Integration Algebra) establishes four identities valid for every smooth y: the Power Portal Formula; the x-Weighted Reduction; the Telescoping Formula for ∫y^(n)y^(m)dx (proved by induction); and the Parity Resolvability Criterion: ∫y^(n)y^(m)dx is expressible in closed form if and only if |n−m| is odd, for y differentially transcendental over Q(x) (proved in both directions). Layer II (Portal Structures) addresses functions whose first antiderivative is non-elementary. Writing y = ∫f dx (non-elementary), the higher iterated antiderivatives I^n = ∫^n f dx^n (n ≥ 2) lie in a finite-dimensional module — the portal. No elementary formula for ∫f dx itself is claimed. Complete verified portal formulas are derived for: e^(−x²) (Gaussian portal, generating function ΣA_n s^n = se^(xs+s²/4)), sin(x)/x (portal dimension 3), √(1+x³), x^x, the complete x^k ln x family, and a new Type TR portal for 1/(x ln x). Additionally, the Differential Galois conjecture is proved: the Picard–Vessiot extension of ∫y^(n)y^(m)dx over the differential polynomial ring C[x]{y} has Galois group isomorphic to G_a = (C,+) when |n−m| is even, and trivial group when |n−m| is odd.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20344519 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | An Integration Algebra for Smooth Functions: Telescoping Formulas, Portal Structures, and the Parity Criterion Wohnsiedler, Simon integration algebra elementary integrability portal structure parity criterion telescopic formula Liouville–Risch Gaussian portal iterated antiderivatives differential Galois theory Picard–Vessiot extension differential transcendence x ln x Type TR portal <p>v4 (22 May 2026): Minor cleanup.<br>(1) HAL identifiers removed from companion paper bibliography entry<br> — the HAL deposits were rejected due to affiliation policy and<br> the identifiers are no longer valid. Zenodo DOI is the sole<br> public identifier.<br>(2) Gel'fand–Dikii (1975) and Mikhailov–Novikov–Wang (2007) added<br> to bibliography. Originality Remark updated to acknowledge the<br> classical background of the parity criterion and precisely<br> enumerate new contributions.</p> <p>────────────────────────────────────────</p> <p>We develop a constructive theory of closed-form integration for smooth functions, in two layers. Layer I (Integration Algebra) establishes four identities valid for every smooth y: the Power Portal Formula; the x-Weighted Reduction; the Telescoping Formula for ∫y^(n)y^(m)dx (proved by induction); and the Parity Resolvability Criterion: ∫y^(n)y^(m)dx is expressible in closed form if and only if |n−m| is odd, for y differentially transcendental over Q(x) (proved in both directions). Layer II (Portal Structures) addresses functions whose first antiderivative is non-elementary. Writing y = ∫f dx (non-elementary), the higher iterated antiderivatives I^n = ∫^n f dx^n (n ≥ 2) lie in a finite-dimensional module — the portal. No elementary formula for ∫f dx itself is claimed. Complete verified portal formulas are derived for: e^(−x²) (Gaussian portal, generating function ΣA_n s^n = se^(xs+s²/4)), sin(x)/x (portal dimension 3), √(1+x³), x^x, the complete x^k ln x family, and a new Type TR portal for 1/(x ln x). Additionally, the Differential Galois conjecture is proved: the Picard–Vessiot extension of ∫y^(n)y^(m)dx over the differential polynomial ring C[x]{y} has Galois group isomorphic to G_a = (C,+) when |n−m| is even, and trivial group when |n−m| is odd.</p> |
| title | An Integration Algebra for Smooth Functions: Telescoping Formulas, Portal Structures, and the Parity Criterion |
| topic | integration algebra elementary integrability portal structure parity criterion telescopic formula Liouville–Risch Gaussian portal iterated antiderivatives differential Galois theory Picard–Vessiot extension differential transcendence x ln x Type TR portal |
| url | https://doi.org/10.5281/zenodo.20344519 |