1. Complex Integration

2. 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

3. 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

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

5. 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

6. Basic Properties
Linearity
Sense of reversal
Partitioning path

7. Evaluation of Complex Integral
Proof :
We have

8. Step to Evaluate

9. Examples

10. Examples (cont.)

11. Examples (cont.)

12. 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.

13. Example

14. Example (cont.)