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.
Then:
, i.e.
and
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
every point of
With
, the function f is analytic at
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
. For any point
on the line whose equation is
on this line, every non empty open ball centered at
has points out of the line, therefore
analytic anywhere (see Figure ).
cannot be
Figure: Why a function is nowhere analytic