Final answer:
The integral of tan(z) around the unit disc is 2πi. The correct answer to this question is A).
Step-by-step explanation:
The integral of tan(z) around the unit disc can be calculated using the Residue Theorem. The integral can be expressed as: ∮|z|=1 tan(z) dz. The integral of tan(z) around the unit disc is 2πi, determined by the residue theorem and the single pole of tan(z) within the unit disc.
Since tan(z) has poles at z = π/2 + nπ, where n is an integer, and all of these poles lie inside the unit disc, we can evaluate the integral by summing the residues at these poles.
The residue at a pole of order 1 is given by the limit as z approaches the pole: Res(tan(z), π/2 + nπ) = 1. Since there are an infinite number of poles, the sum of the residues evaluates to 2πi. Therefore, the answer is (A) 2πi.