Analytic functions. Definition  A function f is analytic at zo if the following conditions are fulfilled: 1. f is derivable at z0 ; 2. There exists a neighborhood V of zo such that f is derivable at every point of V . Definition A function which is analytic on the whole of C is called an entire function. Teorema       A polynomial function is analytic at every point of C , i.e. is an entire function. A rational function is analytic at each point of its domain.

Example Let , i.e. (i.e. and ). Then: C-R equations mean here , i.e. Thus, the origin is the only point where f can be derivable and f is nowhere analytic.

Example If , the function f is analytic at every point of With we have Thus: Cauchy-Riemann equations are verified by the first partial derivatives, and these derivatives are continuous functions on their domain. The function is differentiable at every point of its domain The function f is analytic at every point of

Example Let , i.e. and We have: Thus, Cauchy-Rieman equations are verified if, and only if, , i.e. . It follows that can be derivable only on the line whose equation is . For any point on this line, every non empty open ball centered at has points out of the line, therefore cannot be analytic anywhere (see Figure ).

Figure: Why a function is nowhere analytic