Bibliographic record
Abstract
As a matter of fact, the Stirling formula is a very important formula for estimating the size of factorials, which effectively simplifies the calculation of factorials. Based on the convergency theorem, it is very accurate when n is very small, for example, when n=6, the error is only 1.4%. This formula was first discovered by Abraham de Moivre and Stirling, and mathematicians such as provided much proof of it. In addition, there is a famous proof that only relies on ordinary calculus. With this in mind, this paper attempts to independently solve this problem using simple complex analysis methods. To be specific, contour integral, Residue Theorem, complex series will be demonstrated directly and immediately. At the same time, the proof processing will be presented in detail based on the derivations of the formulae. In the meantime, the current limitations will be clarified and the prospects will be proposed according to the analysis. Overall, these results shed light on guiding further exploration of Stirling Formula proofing and applications.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".