Analytic Functions

A complex function is said to be analytic on a region R,  if it is differentiable at every point in R.
The terms " Holomorphic function " are also used for analytic function.

If a complex function is analytic on a region R, it is infinitely differentiable in R. A complex function may fail to be analytic at one or more point through the presence of singularities.

" A complex function that is analytic at all finite points of the complex plane is said to be entire function ."

A single valued function that is analytic in,  all but possibly a discrete subset of its  domain, and at those singularities goes to infinity like a polynomial is called a meromorphic function.