In the global sense of the whole complex plane, for those non-const holomorphic (entire) functions:
In the local sense of the domain, for those holomorphic (meromorphic) functions with an essential singularity, in any neighborhood of that singularity, the functions:
- not removable: must be unbounded.
- Casorati-Weierstrass: must be dense on the plane. ( very strong result! )
- Picard: omit at most 1 point. ( super strong result! )
The former one is Little Picard Theorem, the latter one is Great Picard Theorem.