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

( )f z C

C

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

C

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

C i

( )f z

C

Basic Properties

• Linearity

• Sense of reversal

• Partitioning path

Evaluation of Complex Integral

Proof : We have z x iy

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.

( )f z 0z 1z

Example

Example (cont.)