Abstract
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.
Highlights
One of the most prominent tools in complex analysis is Cauchy’s Residue Theorem
A cycle C is null-homologous in D ⊂ C, if its winding number for all points in C\D vanishes
C is null-homologous in D, if it can be written as a linear combination of closed curves which are contractible in D
Summary
One of the most prominent tools in complex analysis is Cauchy’s Residue Theorem. To state the classical version of this theorem (see, e.g., [1] or [2, Theorem 1, p. 75]) we briefly recall the following notions: A chain is a finite formal linear combination, k. The result is basically a version of the classical formula (2), but with winding number 1/2 for the singularities on the real axis and where the integral on the left-hand side of (2) is interpreted as a Cauchy principal value. Another very recent version of the residue theorem, where poles of order 1 on the piecewise C1 boundary curve γ of an open set are allowed, is discussed in [4, Theorem 1]. We introduce a generalized, noninteger winding number for piecewise C1 cycles C and a general version of the residue theorem which covers all cases of singularities on C.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
