Unified formulas for arbitrary order symbolic derivatives and anti-derivatives of the power-inverse hyperbolic class 1
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We continue on tackling and giving a complete solution to the problem of finding the nth derivative and the nth anti-derivative, where n can be an integer, a fraction, a real, or a symbol, of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified to less general functions. In this work, we consider two subclasses of the power-inverse hyperbolic class. Namely, the power-inverse hyperbolic sine class { f ( x ) : f ( x ) = Σ l j =1 Pj ( x α j )arcsinh(β j x γ j ), α j ∈ C, β j ∈ C\{0},γ j ∈ R\{0}, (1) and the power-inverse hyperbolic cosine class { f ( x ) : f ( x ) = Σ l j =1 Pj ( x α j )arccosh(β j x γ j ), α j ∈ C, β j ∈ C\{0},γ j ∈ R\{0}, (2) where pj's are polynomials of certain degrees. One of the key points in this work is that the approach does not depend on integration techniques The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration. The motivation of this work comes from the area of symbolic computation. The idea is that: Given a function f in a variable x , can CAS find a formula for the n th derivative, the n th anti-derivative, or both of f ? This enhances the power of integration and differentiation of CAS. In Maple, the formulas correspond to invoking the commands diff( f ( x ) for the n th derivative and int( f ( x ), x$n ) for the n th anti-derivative. A software exhibition will be given using Maple. Example: A unified formula for arcsinh(√ x ) in terms of the Meijer G-function (arcsinh(√ x )) (n) = x (1/2-- n over2√π G 1,2 over 1,2 (1/2,1/2over0, n --1/2│ x ) , │ x │ < 1. (3). The above G-function reduces to the original function if n = 0. It gives derivatives of any order if n > 0 and anti-derivatives of any order if n < 0.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it