# Vocabulary question: singularity for an analytic map

I have a question that is purely on vocabulary. My native language is not english, so I would like to know the usual convention for the following.

When people say "let $f: X \to Y$ be an analytic map", where $X,Y$ are, say, complex manifolds, do they allow for singularities on $X$ ? If yes, what about essential singularities?

For example, consider the following statements:

1. All analytic maps $f: \mathbb{P}^1(\mathbb{C}) \to \mathbb C$ are constants.

2. The exponential function $\exp: \mathbb{P}^1(\mathbb{C}) \to \mathbb C$ is analytic.

Which of those statements would be true, according to usual convention? i.e. can we say that the exponential function is analytic on the projective line, but with an essential singularity ?