We study the resurgence properties of the coefficients $C_n(tau)$ appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of $C_n(tau)$ as $nto +infty$ depends on the parity of $n$. Both $C_{2n-1}(tau)$ and $C_{2n}(tau)$ exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again $C_{2k-1}(tau)$ and $C_{2k}(tau)$, respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, prior to this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.
{"title":"Resurgence in the Transition Region: The Incomplete Gamma Function","authors":"GergHo Nemes","doi":"10.3842/SIGMA.2024.026","DOIUrl":"https://doi.org/10.3842/SIGMA.2024.026","url":null,"abstract":"We study the resurgence properties of the coefficients $C_n(tau)$ appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of $C_n(tau)$ as $nto +infty$ depends on the parity of $n$. Both $C_{2n-1}(tau)$ and $C_{2n}(tau)$ exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again $C_{2k-1}(tau)$ and $C_{2k}(tau)$, respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, prior to this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.","PeriodicalId":515898,"journal":{"name":"Symmetry, Integrability and Geometry: Methods and Applications","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140482518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
After presenting the general framework of `mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral analysis and music composition via the construction of rhythmic canons tiling the integer line. We then present the probabilistic generalization of Babbitt's result we recently introduced in a paper entitled ''New hexachordal theorems in metric spaces with probability measure'' and illustrate the new approach with original constructions and examples.
{"title":"Taking Music Seriously: on the Dynamics of 'Mathemusical' Research with a Focus on Hexachordal Theorems","authors":"Moreno Andreatta, C. Guichaoua, Nicolas Juillet","doi":"10.3842/sigma.2024.009","DOIUrl":"https://doi.org/10.3842/sigma.2024.009","url":null,"abstract":"After presenting the general framework of `mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral analysis and music composition via the construction of rhythmic canons tiling the integer line. We then present the probabilistic generalization of Babbitt's result we recently introduced in a paper entitled ''New hexachordal theorems in metric spaces with probability measure'' and illustrate the new approach with original constructions and examples.","PeriodicalId":515898,"journal":{"name":"Symmetry, Integrability and Geometry: Methods and Applications","volume":"23 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139597927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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