Integration of the complex function  in the counterclockwise direction, around |z – 1| = 1, is

Concept:

Cauchy’s Theorem:

If f(z) is an analytic function and f’(z) is continuous at each point within and on a closed curve C, then

Cauchy’s Integral Formula:

If f(z) is an analytic function within a closed curve and if a is any point within C, then

Residue Theorem:

If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then

Formula to find residue:

1. If f(z) has a simple pole at z = a, then

2. If f(z) has a pole of order n at z = a, then

Application:

Given function is 

Poles: z = 1, -1

|z – 1| = 1

⇒ |x – 1 + iy| = 1

The given region is a circle with the centre at (1, 0) and the radius is 1.

Only pole z = 1, lies within the given region.

Residue at z = 1 is, 

The value of the integral = 2πi × 0.5 = πi