The Mellin Transform of Logarithmic and Rational Quotient Function in terms of the Lerch Function
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
Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mstyle displaystyle="true"> <a:mrow> <a:msubsup> <a:mo stretchy="false">∫</a:mo> <a:mn>0</a:mn> <a:mi>∞</a:mi> </a:msubsup> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:msup> <a:mrow> <a:mi>x</a:mi> </a:mrow> <a:mrow> <a:mi>m</a:mi> <a:mo>−</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msup> <a:msup> <a:mrow> <a:mi mathvariant="normal">log</a:mi> </a:mrow> <a:mrow> <a:mi>k</a:mi> </a:mrow> </a:msup> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>a</a:mi> <a:mi>x</a:mi> </a:mrow> </a:mfenced> </a:mrow> </a:mfenced> <a:mo>/</a:mo> <a:mrow> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:msup> <a:mrow> <a:mi>β</a:mi> </a:mrow> <a:mrow> <a:mn>2</a:mn> </a:mrow> </a:msup> <a:mo>+</a:mo> <a:msup> <a:mrow> <a:mi>x</a:mi> </a:mrow> <a:mrow> <a:mn>2</a:mn> </a:mrow> </a:msup> </a:mrow> </a:mfenced> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>γ</a:mi> <a:mo>+</a:mo> <a:mi>x</a:mi> </a:mrow> </a:mfenced> </a:mrow> </a:mfenced> </a:mrow> </a:mrow> </a:mrow> </a:mrow> </a:mfenced> <a:mtext>d</a:mtext> <a:mi>x</a:mi> </a:mrow> </a:mrow> </a:mstyle> </a:math> , where the parameters <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M2"> <x:mi>a</x:mi> <x:mo>,</x:mo> <x:mi>k</x:mi> <x:mo>,</x:mo> <x:mi>β</x:mi> </x:math> , and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M3"> <z:mi>γ</z:mi> </z:math> are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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