Practice midterm 2

As usual, be sure to include your method, and remember to write your name.

Problem 1. Let $\gamma$ be the circle of radius $3$ centered at the origin. Compute

$${\int }_{\gamma }\frac{z-1}{z^2}𝑑z$$

via parametrization.

The above problem is directed towards Objective 4 (complex integration).

Problem 2. Consider the path $\gamma$ given by the straight line segment from $-i$ to $i+1$.

Compute

$${\int }_{\gamma }z{e}^{z^2}𝑑z.$$

The above problem is directed towards Objective 4 (complex integration).

Problem 3. Let $R$ be the rectangle with corners at $-2-i$, $-2+i$, $2-i$, $2+i$:

Compute

$${\int }_{\partial R}\frac{z^2+1}{z^4-z^3}𝑑z.$$

Hint/timesaver: you may find the formulas $g^{\prime }(z)=1-\frac{2}{{(z-1)}^2}$ and $g^{\prime \prime }(z)=\frac{4}{{(z-1)}^3}$ useful, where $g(z)=\frac{z^2+1}{z-1}$.

The above problem is directed towards Objective 5 (Cauchy’s integral theorem and formula).

Problem 4. Let $f:D\to ℂ$ be a smooth function. For $z_0\in D$, let $g(z)=f(z)-f(z_0)$. Show that

$${\int }_{\partial D}g(z)𝑑z={\int }_{\partial D}f(z)𝑑z.$$

The above problem is directed towards Objective 5 (Cauchy’s integral theorem and formula).

Problem 5. Suppose that $f:ℂ\to ℂ$ is an analytic function such that $|f(z)|\le {|z|}^n$ for all $z\in ℂ$. Conclude that $|f^{(n)}(0)|\le n!$.

The above problem is directed towards Objective 6 (consequences of Cauchy’s theorem and formula).

Problem 6. Let $D$ be the disk of radius $1$ centered at the origin. Verify Pompeiu’s formula for $f(z)={|z|}^2$ at $z_0=0$.

The above problem is directed towards Objective 6 (consequences of Cauchy’s theorem and formula).