Holomorphic

Another name for analytic -- complex-differentiable throughout an open neighborhood of every point in a set.

Holomorphic is the term many complex-analysis texts (and this course) use interchangeably with analytic: a function is holomorphic on an open set if it has a complex derivative at every point of that set. There is no difference in meaning between the two words as used here; which one a given textbook prefers is mostly a matter of tradition.

Examples

Every polynomial in $z$ is holomorphic on the entire complex plane.
" $f$ is holomorphic on the domain $D$ " and " $f$ is analytic on the domain $D$ " say exactly the same thing.