$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}}$

Not all examples should be covered.

For Information

**THEOREM 1 (Cauchy's Theorem)** Suppose that $f$ is analytic on a domain $D$. Let $\gamma$ be a piecewise smooth simple closed curve in $D$ whose inside $\Omega$ also lies in $D$. Then
$$\int_\gamma f(z)\,dz = 0.$$

**DEFINITION** A domain $D$ is *simply-connected* if, whenever $\gamma$ is a simple closed curve in $D$, the inside of $\gamma$ is also a subset of $D$.

**THEOREM 2** Let $D$ be a simply-connected domain and $\Gamma$ a closed curve in $D$ that is composed of a finite number of horizontal and vertical line segments. If $f$ is analytic in $D$, then
$$
\int_\Gamma f(\zeta)\,d\zeta=0.
$$

**THEOREM 3** If $f$ is analytic in a simply-connected domain $D$, then there is an analytic function $F$ on $D$ with $F' = f$ throughout $D$.

**COROLLARY** Let $f$ be analytic on a simply-connected domain $D$, and let $\gamma$ be a piecewise smooth closed curve in $D$. Then
$$\int_\gamma f(z)\,dz = 0.$$

**THEOREM 4 (Cauchy's Formula)**
Suppose that $f$ is analytic on a domain $D$ and that $\gamma$ is a piecewise smooth, positively oriented simple closed curve in $D$ whose inside $\Omega$ also lies in $D$. Then
$$
\int _\gamma \frac{f(\zeta)\, d\zeta}{\zeta -z}=
\left\{\begin{aligned}
& 2\pi if(z) && z\in \Omega,\\
&0 &&z\notin \Omega\cup\gamma.
\end{aligned}\right.
$$

- Evaluate the given integral using Cauchy’s Formula or Theorem. $$\int_{|z|=1} \frac{z\,dz}{(z-2)^2}.$$
- Evaluate the definite trigonometric integral making use of the technique of Examples 6 and 7 in this section. $$ \int_0^{2\pi}\frac{d\theta}{2+\cos(\theta)}. $$
- Evaluate the given integral using the technique of Example 10; indicate which theorem or method you used to obtain your answer.
$$
\int_\gamma \frac{dz}{z^2},
$$
where $\gamma$ is any curve in $\Re z > 0$ joining $1-i$ to $1 + i$.

### Term Test 1