Relation of the Cyclotomic Equation with the Harmonic and Derived Series
We associate some (old) convergent series related to definite integrals with the cyclotomic equation xm-1=0, for several natural numbers m; for example, for m=3, x3-1=(x-1)(1+x+x2) leads to ∫01dx(1/(1+x+x2))=π/(33)=(1-1/2)+(1/4-1/5)+(1/7-1/8)+⋯. In some cases, we express the results in terms of the...
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| Format: | Article |
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Wiley
2015-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2015/950521 |
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| author | Luis J. Boya Cristian Rivera |
| author_facet | Luis J. Boya Cristian Rivera |
| author_sort | Luis J. Boya |
| collection | DOAJ |
| description | We associate some (old) convergent series related to definite integrals with the cyclotomic equation xm-1=0, for several natural numbers m; for example, for m=3, x3-1=(x-1)(1+x+x2) leads to ∫01dx(1/(1+x+x2))=π/(33)=(1-1/2)+(1/4-1/5)+(1/7-1/8)+⋯. In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved. |
| format | Article |
| id | doaj-art-23f42da56d3e447cb2ede0c74eafec25 |
| institution | OA Journals |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-23f42da56d3e447cb2ede0c74eafec252025-08-20T02:04:14ZengWileyThe Scientific World Journal2356-61401537-744X2015-01-01201510.1155/2015/950521950521Relation of the Cyclotomic Equation with the Harmonic and Derived SeriesLuis J. Boya0Cristian Rivera1Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, SpainDepartamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, SpainWe associate some (old) convergent series related to definite integrals with the cyclotomic equation xm-1=0, for several natural numbers m; for example, for m=3, x3-1=(x-1)(1+x+x2) leads to ∫01dx(1/(1+x+x2))=π/(33)=(1-1/2)+(1/4-1/5)+(1/7-1/8)+⋯. In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved.http://dx.doi.org/10.1155/2015/950521 |
| spellingShingle | Luis J. Boya Cristian Rivera Relation of the Cyclotomic Equation with the Harmonic and Derived Series The Scientific World Journal |
| title | Relation of the Cyclotomic Equation with the Harmonic and Derived Series |
| title_full | Relation of the Cyclotomic Equation with the Harmonic and Derived Series |
| title_fullStr | Relation of the Cyclotomic Equation with the Harmonic and Derived Series |
| title_full_unstemmed | Relation of the Cyclotomic Equation with the Harmonic and Derived Series |
| title_short | Relation of the Cyclotomic Equation with the Harmonic and Derived Series |
| title_sort | relation of the cyclotomic equation with the harmonic and derived series |
| url | http://dx.doi.org/10.1155/2015/950521 |
| work_keys_str_mv | AT luisjboya relationofthecyclotomicequationwiththeharmonicandderivedseries AT cristianrivera relationofthecyclotomicequationwiththeharmonicandderivedseries |