Analytic functions.

Definition 3.5.1   A function $ f$ is analytic at $ z_0$ if the following conditions are fulfilled:
  1. $ f$ is derivable at $ z_0$ ;
  2. There exists a neighborhood $ V$ of $ z_0$ such that $ f$ is derivable at every point of $ V$ .

The simplest open neighborhood of a point $ z_0$ is an open ball centered at $ z_0$ ; we will generally use such balls.

Definition 3.5.2   Let $ f$ be a function defined over an open subset $ V$ of $ \mathbb{C}$ . the function $ f$ is analytic over $ V$ if it is analytic at each point of $ V$ .

A function which is analytic over the whole of $ \mathbb{C}$ is called an entire function.

Proposition 3.5.3       

  1. A polynomial function is analytic at every point of $ \mathbb{C}$ , 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.
$ \qedsymbol$

Example 3.5.4   Let $ f(z)=\vert z\vert^2$ , i.e. $ f(z)=x^2+y^2$ (i.e. $ u(x,y)=x^2+y^2$ and $ v(x,y)=0$ ). Then:

$\displaystyle u_x=2x, \; u_y=2y, \; v_x=v_y=0.$    

C-R equations mean here \begin{displaymath}\begin{cases}2x=0 \ 2y=0\end{cases}\end{displaymath} , i.e. $ (x,y)=(0,0)$ . Thus, the origin is the only point where $ f$ can be derivable and $ f$ is nowhere analytic.

Example 3.5.5   We use the function and the results of Example 3.5. For any point $ z_0$ on one of these axes, any open ball centered at $ z_0$ contains points where $ f$ is not differentiable. Thus, $ f$ is not analytic at any point.

Example 3.5.6   If $ f(z)=\frac {1}{z^2+1}$ , the function $ f$ is analytic at every point of $ \mathbb{C} - \{ -i, i \}$ :

Example 3.5.7   Let $ f(z)=x^2+iy^2$ , i.e. $ U(x,y)=x$ and $ v(x,y)=y$ . We have:

\begin{displaymath}\begin{cases}u_x=2x \ u_y=0 \end{cases} \qquad \text{and} \qquad \begin{cases}v_x=0 \ v_y=2y \end{cases}\end{displaymath}    

Thus, Cauchy-Rieman equations are verified if, and only if, $ 2x=2y$ , i.e. $ x=y$ . It follows that $ f$ can be derivable only on the line whose equation is $ x-y=0$ . For any point $ z_0$ on this line, every non empty open ball centered at $ z_0$ has points out of the line, therefore $ f$ cannot be analytic anywhere (see Figure 3).

Figure 3: Why a function is nowhere analytic.
\begin{figure}\mbox{\epsfig{file=NonAnalyticOnLine.eps,height=4cm}}\end{figure}

Noah Dana-Picard 2007-12-24