Final answer:
To integrate 9tan(z) / z^4 counterclockwise around the curve C: |z|=1, we can use the residue theorem. First, we find the residues of the function at its singular points inside the curve. Finally, we evaluate the integral using the residue theorem.
Step-by-step explanation:
To integrate 9tan(z) / z^4 counterclockwise around the curve C: |z|=1, we can use the residue theorem. First, we need to find the residues of the function at its singular points inside the curve. The function has a simple pole at z=0. We can calculate the residue by finding the coefficient of the (z-0)^3 term in the Laurent series expansion of the function.
After finding the residue, we can use the residue theorem to evaluate the integral. The integral around the curve C is equal to 2πi times the sum of the residues of the function at its singular points inside the curve.
Substituting the value of the residue into the formula, we get ∮c 9tanz / z⁴ dz = 2πi * residue = 2πi * (coefficient of (z-0)^3 term in the Laurent series expansion) = ... (exact answer).