The simplest open neighborhood of a point
is an open ball centered at
; we will generally use such balls.
be a function defined over an open subset
. 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.
- A polynomial function is analytic at every point of
, i.e. is an entire function.
- A rational function is analytic at each point of its domain.
- A polynomial function is an entire function, as a consequence of Proposition 2.4 and Ex. 2.3.
- We use once again Proposition 2.4, together with the previous alinea,
as a rational function is the quotient of two polynomial functions.
C-R equations mean here
. Thus, the origin is the only point where
be derivable and
is nowhere analytic.
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.
, the function
is analytic at every
- With the notations of 3.1, we have:
- 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
differentiable at every point of its domain
is not defined at
is not entire.
. We have:
Thus, Cauchy-Rieman equations are verified if, and only if,
. 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
Why a function is nowhere analytic.