Derivation of Some Entries in the Tables of David Bierens De Haan and Anatolii Prudnikov: An Exercise in Integration Theory
Bibliographic record
Abstract
It is always useful to improve the catalogue of definite integrals available in tables. In this paper we use our previous work on Lobachevsky integrals to derive entries in the tables by Bierens De Haan and Anatolli Prudnikov featuring errata results and new integral formula for interested readers. In this work we derive a definite integral given by (1) in terms of the Lerch function. The importance of this work lies in the derivation of known and new results not presently found in current literature. We used our contour integral method and applied it to an integral in Prudnikov and used it to derive a closed form solution in terms of a special function. The advantage of using a special function is the added benefit of analytic continuation which widens the range of computation of the parameters. Special functions have significance in mathematical analysis, functional analysis, geometry, physics, and other applications. Special functions are used in the solutions of differential equations or integrals of elementary functions. Special functions are linked to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.037 | 0.019 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".