Complex analysis & residues

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Let $C$ be a positively oriented simple closed contour enclosing the point $z_0 = -2$, and let $f(z) = 2z^{2} + 3z + 2$, which is entire. Using Cauchy's Integral Formula, evaluate $\dfrac{1}{2\pi i}\oint_C \dfrac{f(z)}{z - z_0}\,dz$.

2. Warm-up

   Let $C$ be a positively oriented simple closed contour enclosing the point $z_0 = -3$, and let $f(z) = 3z^{2} + 3z + 1$, which is entire. Using Cauchy's Integral Formula, evaluate $\dfrac{1}{2\pi i}\oint_C \dfrac{f(z)}{z - z_0}\,dz$.

3. Warm-up

   Let $C$ be a positively oriented simple closed contour enclosing the point $z_0 = 3$, and let $f(z) = 5z^{2} - 3z - 3$, which is entire. Using Cauchy's Integral Formula, evaluate $\dfrac{1}{2\pi i}\oint_C \dfrac{f(z)}{z - z_0}\,dz$.

Problems shown under their original terms: Generated (template, solver-computed).