Non-Integer Valued Winding Numbers and a Generalized Residue Theorem
We define a generalization of the winding number of a piecewise C1 cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value but is also possible in a real version via an integral with...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2019/6130464 |
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| author | Norbert Hungerbühler Micha Wasem |
| author_facet | Norbert Hungerbühler Micha Wasem |
| author_sort | Norbert Hungerbühler |
| collection | DOAJ |
| description | We define a generalization of the winding number of a piecewise C1 cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value but is also possible in a real version via an integral with bounded integrand. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply. |
| format | Article |
| id | doaj-art-940e15b370234f58ba55b4cb35b2f3a2 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-940e15b370234f58ba55b4cb35b2f3a22025-02-03T01:03:35ZengWileyJournal of Mathematics2314-46292314-47852019-01-01201910.1155/2019/61304646130464Non-Integer Valued Winding Numbers and a Generalized Residue TheoremNorbert Hungerbühler0Micha Wasem1Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, SwitzerlandHTA/HSW Freiburg, HES-SO University of Applied Sciences and Arts Western Switzerland, Pérolles 80/Chemin du Musée 4, 1700 Freiburg, SwitzerlandWe define a generalization of the winding number of a piecewise C1 cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value but is also possible in a real version via an integral with bounded integrand. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply.http://dx.doi.org/10.1155/2019/6130464 |
| spellingShingle | Norbert Hungerbühler Micha Wasem Non-Integer Valued Winding Numbers and a Generalized Residue Theorem Journal of Mathematics |
| title | Non-Integer Valued Winding Numbers and a Generalized Residue Theorem |
| title_full | Non-Integer Valued Winding Numbers and a Generalized Residue Theorem |
| title_fullStr | Non-Integer Valued Winding Numbers and a Generalized Residue Theorem |
| title_full_unstemmed | Non-Integer Valued Winding Numbers and a Generalized Residue Theorem |
| title_short | Non-Integer Valued Winding Numbers and a Generalized Residue Theorem |
| title_sort | non integer valued winding numbers and a generalized residue theorem |
| url | http://dx.doi.org/10.1155/2019/6130464 |
| work_keys_str_mv | AT norberthungerbuhler nonintegervaluedwindingnumbersandageneralizedresiduetheorem AT michawasem nonintegervaluedwindingnumbersandageneralizedresiduetheorem |