I find interesting how often in complex analysis the Cauchy theorem is used in undegraduate exercices, as opposed to the rarity of usage seen for it's "complementary" Morera's theorem.
An interesting application would be to find the domain of holomorphy of the composition of a certain known function $f(z)$ which is holomorfic in a domain $D \subseteq \mathbb{C}$
For instance, we know that the function that maps a complex number $z$ to its conjugate $z^*$ is not holomorphic, but it would be interesting to find the domain of the function: $$f^*(z^*)$$ using Morera's theorem.
Maybe it could be done by using other tools with a more specific function, but I am confident that Morera's theorem is a good approach if the function $f(z)$ considered is just a generic holomorfic function.
How could it be done?