Cauchy theorem: If a function $f$ is analytic on a simply connected domain $D$ and $C$ is a simple closed contour lying in $D$ then $ \int _{\gamma} f(z)dz=0 $.
My doubt Cauchy's theorem states that a function $f$ must be analytic in a domain $D$ and $\gamma$ is a closed contour within $D$. However, I often encounter questions that mention $\gamma$ but doesn't specify the domain $D$. Many questions do not explicitly mention the domain, but assumption is typically that you should chhose an appropriate $D$ to make theorem applicable.
For example using this theorem $ \int _{\gamma} \frac{1}{z^2}dz=0 $, where ${\gamma}$ is $(x-2)^2 + \frac{1}{4}(y - 5)^2 = 1$. My confusion is what domain $D$ should I consider where I can see $\gamma$ is a closed contour. Should I take arbitrary domain containing $\gamma$?
Kindly help me to clear this doubt.