Final answer:
The value of the integral of g(z) around the circle |z-i|=2 is determined by computing the residues of the functions at poles within the circle and applying the Cauchy's residue theorem.
Step-by-step explanation:
The student has asked to find the value of the integral of g(z) around the circle |z-i|=2 in the positive sense for the two functions g(z) = 1/z²+4 and g(z) = 1/(z²+4)². To evaluate these integrals, we can use the Cauchy's residue theorem if the poles of these functions are within the given circle. In case (a), the function has poles at z = 2i and z = -2i, but only z = 2i is within the circle |z-i|=2. We can calculate the residue at this pole and use the residue theorem to find the integral. In case (b), we need to find the residue of the function at its pole within the circle, which again is z = 2i, and apply the residue theorem.