×

Non-integer valued winding numbers and a generalized residue theorem. (English) Zbl 1450.30059

Summary: We define a generalization of the winding number of a piecewise \(C^1\) 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.

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
53A04 Curves in Euclidean and related spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlfors, L. V., Complex Analysis. Complex Analysis, An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (1978), New York, NY, USA: McGraw-Hill Book Co., New York, NY, USA
[2] Narasimhan, R.; Nievergelt, Y., Complex Analysis in One Variable (2001), Birkhäuser Boston · Zbl 1009.30001
[3] Henrici, P., Applied and Computational Complex Analysis. Applied and Computational Complex Analysis, Power series—integration—conformal mapping—location of zeros, Pure and Applied Mathematics, 1 (1974), New York, NY, USA: Wiley-Interscience [John Wiley & Sons], New York, NY, USA · Zbl 0313.30001
[4] Legua, M.; Sánchez-Ruiz, L. M., Cauchy principal value contour integral with applications, Entropy, 19, 5 (2017)
[5] Lu, J. K., A generalized residue theorem and its applications, Wuhan Daxue Xuebao, 3, 1-8 (1978)
[6] Zhong, S. G., A generalized residue theorem for unbounded multiply connected regions of the second class, Acta Mathematica Scientia, 14, 2, 163-167 (1994) · Zbl 0900.30044
[7] Zhong, S. G., Transcendental singular integrals and a generalized residue theorem, Acta Mathematica Scientia, 15, 1, 43-48 (1995) · Zbl 0900.30045
[8] Baker, J. A., Plane curves, polar coordinates and winding numbers, Mathematics Magazine, 64, 2, 75-91 (1991) · Zbl 0742.30001
[9] Yin, C.; Jiang, H.; Li, L.; Lü, R.; Chen, S., Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems, Physical Review A: Atomic, Molecular and Optical Physics, 97, 5 (2018)
[10] Hazewinkel, M., Encyclopaedia of Mathematics: Reaction-Diffusion Equation - Stirling Interpolation Formula (2012), Springer Netherlands
[11] Lavrentyev, M. A.; Shabat, B. V., Methods of the Theory of Complex Variable Functions (M. L.: GITTL, p. 606, 1951), Moscow: Publishing House Science, Moscow
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.