Relationship between analytic and holomorphic

Question

Is every analytic function holomorphic? But not every holomorphic function is analytic?

Answer 1

For functions of a single complex variable, a function being analytic is equivalent to it being holomorphic. That is, Analytic $\implies$ Holomorphic, and Holomorphic $\implies$ Analytic.

Up till recently, I would have agreed with P Vanchinathan's comment, that the two are merely synonyms of each other, but I no longer think that's the best way of looking at it. Of course these are merely words, and so one can define them however one wants, but a good way (and from what I understand, a fairly standard way) to define them is a complex function being "analytic" at a point means that it has a convergent Taylor Series on some open disk centered at that point. In contrast, a function being "holomorphic" at a point simply means that it is differentiable on some open disk centered at that point. With these two definitions they are no longer mere synonyms, but (amazingly) they are equivalent. I think Wikipedia says it well:

Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighborhood of each point in its domain. The fact that all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.