A Note on Some Integrals by Malmsten and Bierens de Haan
Bibliographic record
Abstract
Carl Johan Malmsten (1846) and David Beirens de Haan (1847) published work containing some interesting integrals. While no formal derivations of the integrals in his book Nouvelles Tables d'Intègrales Dèfines are available in current literature deriving and evaluating such formulae are useful in all aspects of science and engineering whenever such formulae are used. Formulae in the book of Bierens de Haan are used in connection with certain potential problems where there is the need to determine the vector potential of two parallel, infinitely long, tubular rectangular conductors carrying cur-rents in opposite directions. In this current work we supply formal derivations for some of these integrals along with deriving some special cases as new integrals in order to expand upon the book of Bierens de haan to aid in potential research where these formulae are applicable. Updating book of integrals is always a useful exercise as it keeps the volume accurate and more useful for potential readers and researchers. Formal derivations are also useful as they help in verifying the correctness of integrals in such volumes. The definite integral we derived in this work is given by (1) in terms of the Lerch function, where the parameters a; k; m; and p are general complex numbers subject to their restrictions. This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.
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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 | 0.004 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.004 | 0.007 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.005 | 0.004 |
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".