What are the difference among entire, smooth, analytic and holomorphic in complex function?

Question

In complex analysis, I know that they have similar meaning.
However what are the difference among them? Especially for entire function, does entire function mean analytic function?

Answer 1

"$f$ is holomorphic on $U$" means "the complex derivative of $f$ exists and is continuous on $U$".

"$f$ is entire" means means "$f$ is holomorphic on $\mathbb{C}$".

"$f$ is smooth on $U$" means a lot of different things in different contexts. For complex analysis, let us introduce the map $p : \mathbb{C} \to \mathbb{R}^2$ given by $p(x+iy)=(x,y)$. Then "$f$ is smooth on $U$" probably means that $p \circ f \circ p^{-1} : \mathbb{R}^2 \to \mathbb{R}^2$ is infinitely differentiable on $p(U)$. (In other words, the underlying function on the plane, where we forget about complex numbers, is smooth.) This concept doesn't come up very often in complex analysis because you really need to work with holomorphic functions to use complex-analytic methods.

"$f$ is analytic on $U$" a priori means "a power series expansion of $f$ exists on $U$". It is a remarkable fact in complex analysis that this is equivalent to $f$ being holomorphic on $U$. As a result of this the two terms are sometimes used interchangeably. Another result of this is that holomorphic functions are smooth (in the sense of the previous paragraph).