Definition of Analytic Functions in Complex Analysis

[thoughtgaze]

In Mathematics of Classical and Quantum Mechanics by Byron and Fuller, they state that "Some authors (never mathematicians) define an analytic function as a differentiable function with a continuous derivative." ..."But this is a mathematical fraud of cosmic proportions.. "

Their main point is that you don't have to assume continuity of the first derivatives of an analytic function to prove Cauchy's Integral Theorem if you use the Goursat approach, yet I thought that really IS how an analytic function is defined, i.e. that a function of a complex variable is analytic within a region S if it is differentiable within and on the boundary of S.

 

[lurflurf]

The confusion as to the definition of analytic results from there being several equivalent definitions. There is the complex differentiable once, the power series one, the Cauchy–Riemann one, the path independent one, and so forth. The reason to require continuity is it makes for easier proofs, in fact by Looman Menchoff theorem only Fréchet differentiability is required. The main point is that there is no one definition everyone agrees on, but they are mostly talking about the same thing. This makes for comical understandings when two people disagree on the definition.