Analytic functions.

Definition 3.5.1   A function is analytic at if the following conditions are fulfilled:
1. is derivable at ;
2. There exists a neighborhood of such that is derivable at every point of .

The simplest open neighborhood of a point is an open ball centered at ; we will generally use such balls.

Definition 3.5.2   Let be a function defined over an open subset of . the function is analytic over if it is analytic at each point of .

A function which is analytic over the whole of is called an entire function.

Proposition 3.5.3

1. A polynomial function is analytic at every point of , i.e. is an entire function.
2. A rational function is analytic at each point of its domain.

Proof.

1. A polynomial function is an entire function, as a consequence of Proposition 2.4 and Ex. 2.3.
2. We use once again Proposition 2.4, together with the previous alinea, as a rational function is the quotient of two polynomial functions.

Example 3.5.4   Let , i.e. (i.e. and ). Then:

C-R equations mean here , i.e. . Thus, the origin is the only point where can be derivable and is nowhere analytic.

Example 3.5.5   We use the function and the results of Example 3.5. For any point on one of these axes, any open ball centered at contains points where is not differentiable. Thus, is not analytic at any point.

Example 3.5.6   If , the function is analytic at every point of :
• With the notations of 3.1, we have:

Thus:

• Cauchy-Riemann equations are verified by the first partial derivatives, and these derivatives are continuous functions on their domain. By Thm 3.4, the function is differentiable at every point of its domain .
• As is not defined at and at , is not entire.

Example 3.5.7   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 3).

Noah Dana-Picard 2007-12-24