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

Yes, that is correct! Good job using the argument principle to find the number of poles and zeros inside the contour. Then, applying the residue theorem, the value of the integral is equal to -2πi times the residue at the pole, which in this case is located at z = π/2 - π/4 = π/4. Therefore, the integral is 2πi.