Complex Number Problems

Problem 1. Find the Laurent series expansion off(z) = 1/ z(z + 1)in the region 0 < |z| < 1. Problem 2. Determine the Laurent series expansion of the functionf(z) = 1 /z(1 + z2)(4 − z2)in the regionsa) 0 < |z| < 1, b) 1 < |z| < 2, c) |z| > 2. Problem 3. Determine the Laurent series expansion of the functionf(z) = e^(1/z) + e^zin the region |z| > 0. Problem 4. Compute the residue of the functionf(z) = (1 + z^6)^(−1) at the point z = −i.Problem 5. Compute by contour integrationx^2/(1 + x2)^2dx from 0 to ∞