Complex Integrals Using Argument Principle

Question

Integrate tan z dz along the counterclockwise direction contour C:|z-(pi/2)|=1, using argument principle

Here, I get

no. of poles inside C = 1

No. Of zeros inside C = 0

Thus, the answer for the integral as =-2pi*(i)(0-1)=2*pi*I

Is this correct?

Answer 1

According to the argument principle, the integral can be calculated as:

∮c tan(z) dz = 2πi (no. of poles inside - no. of zeros inside)

In this case, the integral becomes:

∮c tan(z) dz = 2πi (1 - 0) = 2πi

Therefore, the correct answer for the integral is 2πi. You are correct!