Complex Analysis – Difference Between Analytic and Holomorphic Functions

Question

A function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be holomorphic in an open set $A \subset \mathbb{C}$ if it is differentiable at each point of the set $A$.

The function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be analytic if it has power series representation.

We can prove that the two concepts are same for a single variable complex functions. So why these two different terms? Is there any difference between these two concepts in general, please give example.

Thank you for your help.

Answer 1

So why these two different terms?

Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:

In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.

Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.

Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:

[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."

As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.