Definition 3.2.1
Let
be a function defined on a domain
in
and let
be an interior point of
. The function
is
derivable (or
differentiable) at
if there
exists a complex number
such that
We denote this by
.
Remark 3.2.2
Another formulation of Def.
2.1 is as follows:
Example 3.2.3
Let
, where
. Then, for every
:
from what follows:
The function
is differentiable at every point
and
.
The algebra of derivable functions of a complex variable is (formally) identical to
the algebra of derivable functions of a real variable seen in Calculus.
Example 3.2.5
If
, then
is differentiable at every point of
and
.
As easy consequences of Proposition 2.4, we have Corollary 2.7 and
Corollary 2.8.
Corollary 3.2.7
A polynomial function is differentiable on the whole of
.
Corollary 3.2.8
A rational function is differentiable at every point of its domain of definition
Example 3.2.9
The function
such that
is differentiable on
and we have:
Actually, we do not dispose of enough functions in order to display more interseting examples. For that purpose, we shall wait until next chapter (i.e. Chapter chapter analytic functions).
Noah Dana-Picard
2007-12-24