EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I
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Bibliographic record
Abstract
The non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le2\beta+1\), where \(\{\beta,\alpha\}\in\mathbb{R}\), are evaluated in terms of the hypergeometric functions \(_{1}F_2\) and \(_{2}F_3\), and their asymptotic expressions for \(|x|\gg1\) are also derived. The integrals of the form \(\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\) and \(\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\), where \(n\) is a positive integer, are expressed in terms \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\), and then evaluated. \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\) are also evaluated in terms of the hypergeometric function \(_{2}F_2\). And so, the hypergeometric functions, \(_{1}F_2\) and \(_{2}F_3\), are expressed in terms of \(_{2}F_2\). The exponential integral \(\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx\) where \(\beta\ge1\) and \(\alpha\le\beta+1\) and the logarithmic integral \(\text{Li}=\int_{\mu}^{x} dt/\ln{t}\), \(\mu>1\), are also expressed in terms of \(_{2}F_2\), and their asymptotic expressions are investigated. For instance, it is found that for \(x\gg2\), \(\text{Li}\sim {x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\) is added to the known expression in mathematical literature \(\text{Li}\sim {x}/{\ln{x}}\). The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given.
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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 | 0.007 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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