非整值卷绕数与广义留数定理
Non-integer valued winding numbers and a generalized Residue Theorem
摘要：
我们定义了复平面中分段$C^1$圈的卷绕数的推广，它对于位于圈上的点也具有几何意义。这个卷绕数的计算依赖于柯西主值，但也可以通过有界被积函数的积分在实版本中进行。新的卷绕数允许建立一个广义的留数定理，它也涵盖的情况下，奇点躺在循环。这个留数定理可以用来计算反常积分的值，对于这些积分，经典的留数定理的标准方法并不适用。
英文摘要：
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.
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