柯西积分公式的推广及应用

The Popularization and Application of Cauchy Integral Formula

当积分曲线内有两个以上奇点时,柯西积分公式及其高阶形式不再适用。通过构造复周线或者重塑被积函数,利用推广的柯西积分公式可以解决具有多个奇点的积分问题。当被积函数在积分曲线内包含多个高阶极点时,利用柯西留数定理建立了高阶柯西积分公式的推广形式;当被积函数在积分曲线外含有一个有限奇点时,柯西积分公式被推广到了无界域上,从而揭示了柯西型积分与该奇点函数值之间的关系。

When there are more than two singular points in the integral curve,the Cauchy integral formula and its higher-order form are not available. At this moment,by constructing the complex closed curve or reconstructing the integrand,the integral problem with several singular points can be solved by using the generalized Cauchy integral formula. When the integrand function contains several higher-order poles within the integral curve,the generalized form of higher-order Cauchy integral formula is established by using Cauchy residue theorem; when the integrand function contains a finite singular point outside the integral curve,the Cauchy integral formula is extended to the unbounded domain,so the relation is revealed between the cauchy-type integral and the function value of the singular point.