Complex Integration

Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is integrated over a given curve . The curve is a path of integration with parametric representation Examples

Line Integral in Complex Plane (cont.) We assume the curve is smooth, thus it has continuous and non-zero derivatives The definition of the derivatives

Definition of Complex Line Integral Suppose that the time in parametric representation is divided into The is indeed partitioned into where

Definition of Complex Line Integral (cont.) On each subdivision of we choose any point The we can form the sum The limit of the sum is termed as line integration of over the path of integration of Important definition

Basic Properties Linearity Sense of reversal Partitioning path

Evaluation of Complex Integral Proof : We have

Step to Evaluate

Examples

Examples (cont.)

Examples (cont.)

Dependence on Path If 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.

Example

Example (cont.)