2Line Integral in Complex Plane A complex definite integrals are called complex line integralsIt denotes asHere, the integrand is integrated over a given curve .The curve is a path of integration with parametric representationExamples
3Line Integral in Complex Plane (cont.) We assume the curve is smooth, thus it has continuous and non-zero derivativesThe definition of the derivatives
4Definition of Complex Line Integral Suppose that the time in parametric representation is divided intoThe is indeed partitioned intowhere
5Definition of Complex Line Integral (cont.) On each subdivision of we choose any pointThe we can form the sumThe limit of the sum is termed as line integration of over the path of integration ofImportant definition
6Basic PropertiesLinearitySense of reversalPartitioning path
12Dependence on PathIf we integrate from to along different paths, then the integral in general will have different values.A complex line integral depends not only on the endpoints of the path, but also on the path itself.