Choose the correct statement(s) from the following options, regarding Cauchy's theorem on complex integration $\oint_{C} f(z) \mathrm{d} z$ where $C$ is a simple closed path in a simply connected domain $D$.
- Cauchy's theorem cannot be directly applied to conclude that $\oint_{C} f(z) \mathrm{d} z=0$ when $f(z)=\frac{1}{z^{2}}$, and $C$ is the unit circle
- If $f(z)$ is analytic in $D$, then it can be concluded that $\oint_{C} f(z) \mathrm{d} z=0$ for any simple closed path $C$ in $D$
- The function $f(z)$ must be analytic in $D$ to conclude $\oint_{C} f(z) \mathrm{d} z=0$ for any simple closed path $C$ in $D$
- $\oint_{C} f(z) \mathrm{d} z \neq 0$ when $f(z)=\frac{1}{z^{2}}$, since the function is not analytic at $z=0$
